# On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows
and ergodicity of their extensions

Krzysztof Frączek  Faculty of Mathematics and Computer Science, Nicolaus
Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
<EMAIL_ADDRESS>and  Corinna Ulcigrai  Institut für Mathematik, Universität
Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
<EMAIL_ADDRESS>
###### Abstract.

We consider smooth area-preserving flows (also known as _locally Hamiltonian
flows_) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth
observables along the flow trajectories. We show that these integrals display
a _power deviation spectrum_ and describe the cocycles that lead the pure
power behaviour, giving a new proof of results by Forni (Annals 2002) and
Bufetov (Annals 2014) and generalizing them to observables which are non-zero
at fixed points. This in particular completes the proof of the original
formulation of the Kontsevitch-Zorich conjecture. Our proof is based on
building suitable _correction operators_ for cocycles with logarithmic
singularities over a full measure set of interval exchange transformations
(IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles
over IETs. In the case of symmetric singularities, exploiting former work of
the second author (Annals 2011), we prove a tightness result for a finite
codimension class of observables. We then apply the latter result to prove the
existence of ergodic infinite extensions for a full measure set of locally
Hamiltonian flows with non-degenerate saddles in any genus $g\geq 2$.

###### 2000 Mathematics Subject Classification:

37E35, 37A40, 37A10, 37C83

##  1\. Introduction and main results

In this paper we give a contribution to the study of ergodic theory of smooth
area-preserving flows on higher genus surfaces (also known as locally
Hamiltonian flows) as well as to the infinite ergodic theory of flow
extensions. The class of surface flows that we work with is introduced in §
1.1. We study in particular _deviations of ergodic averages_ , by proving the
existence of a _power deviation_ spectrum for the ergodic integrals along the
flow. This extends and gives a new proof of results by Forni [23] and Bufetov
[6] for observables with compact support _outside_ a neighbourhood of the
fixed points of the flow, to observables which have full support and are _non-
zero_ at singularities. We then use our result to show the existence of
infinite extensions of such flows which are _ergodic_ with respect to the
natural infinite invariant measure. This result generalizes to higher genus a
classical result by Krygin [42] in genus one and extends a previous result in
higher genus by the authors (see [20], where we showed the existence of
ergodic extensions in any genus, but only for flows with self-similar
foliations) to a full measure set of flows.

###  1.1. Locally Hamiltonian flows

Let $M$ be a compact, connected, orientable (smooth) surface and let $g$
denote its genus. We will assume throughout that $g\geq 1$. We will consider
smooth flows on $M$ preserving a smooth measure $\mu$ (i.e. absolutely
continuous measure with smooth positive density), see § 2.1. These flows, also
known in the literature as _multi-valued Hamiltonian_ , are _locally
Hamiltonian_ flows: indeed, the flow
$\psi_{\mathbb{R}}:=(\psi_{t})_{t\in{\mathbb{R}}}$ is _locally_ Hamiltonian in
the sense that around any point in $M$ one can find coordinates
$(x_{1},x_{2})$ on $M$ in which $\psi_{\mathbb{R}}$ is locally given by the
solution to the equations

$\begin{cases}\dot{x}_{1}&={\partial H}/{\partial x_{2}},\\\
\dot{x}_{2}&=-{\partial H}/{\partial x_{1}}\end{cases}$

for some smooth real-valued Hamiltonian function $H$. A _global_ Hamiltonian
$H$ cannot be in general defined (see [50], § 1.3.4), but one can think of
$\psi_{\mathbb{R}}$ as globally given by a _multi-valued_ Hamiltonian
function. We will assume throughout this paper that the fixed points of
$\psi_{\mathbb{R}}$ are _non-degenerate_ (also called _Morse_ fixed points),
namely that for every fixed point $p$ the local Hamiltonian $H$ is a Morse
function at $p$.

The interest in the study of multi-valued Hamiltonians and the associated
flows in higher genus ($g\geq 1$) and, in particular, in their ergodic and
mixing properties, was highlighted by Novikov [51] in connection with problems
arising in solid-state physics as well as in pseudo-periodic topology (see
e.g. the survey [72] by A. Zorich). The simplest examples of locally
Hamiltonian flows with singularities on a torus, i.e. flows with one center
and one simple saddle (see Figure 1(a)), were studied by V. Arnold in [2] and
are nowadays often called _Arnold flows_ 111More precisely, referring to the
decomposition described in § 2.1.1, we call _Arnold flow_ the restriction to a
minimal component obtained by removing the center and the disk filled by
periodic orbits around it (called _island_), which, as Arnold shows in [2], is
always bounded by a saddle loop..

(a) An Arnold flow ($g=1)$

(b) A flow on a surface of $g=3$

Figure 1. Examples of locally Hamiltonian flows on a surfaces.

On the space of locally Hamiltonian flows, one can define a _topology_ (see §
2.1.1) as well as a measure class (the _Katok fundamental class_ , see §
2.1.2). Our understanding of the _typical_ chaotic properties (in the measure
theoretical sense) of these flows has advanced a lot in the last forty years.
While results concerning orbit properties, such as minimality or ergodicity,
were known first, since they can be simply deduced222One can show (see for
example [72]) that every _minimal_ locally Hamiltonian flow on $M$ (as well as
the restriction of a locally Hamiltonian flow to one of its minimal components
(see § 2.1.1) has the same _trajectories_ (up to time-reparametrization) as a
_translation flow_. Thus, one can infer properties which depend only on
trajectories as sets and not on their time-parametrization, such as minimality
and ergodicity, from the known properties of typical translation flows. from
classical results which were proved using Teichmüller dynamics (see below as
well as § 2.1.4), results on finer chaotic properties such as (weak or
multiple) mixing, or recently spectral and disjointness results, were proved
only in the last twenty years, since they depend on the movement along
trajectories (i.e. on time-reparametrization) and require more delicate
estimates exploiting the locally Hamiltonian parametrization of the orbits. We
summarize some of the known results in § 2.1.4 below.

In the classification of chaotic behaviour in locally Hamiltonian flows it is
crucial to distinguish between two open sets (complementary, up to measure
zero, see § 2.1.2 for more details): in the first open set, which we will
denote by $\mathscr{U}_{min}$, the typical flow is _minimal_ , in the sense
that the orbits of all points which are not fixed points are _dense_ in $M$.
On the other open set, that we call $\mathscr{U}_{\neg min}$, the flow is not
minimal, but one can decompose the surface into a finite number of subsurfaces
with boundary $M_{i}$, $i=1,\dots,N$ such that for each $i$ either $M_{i}$ is
a _periodic component_ , i.e. the interior of $M_{i}$ is foliated into closed
orbits of $\psi_{\mathbb{R}}$ (in Figure 1 (b) one can see three periodic
components, namely two disks and one cylinder, all foliated by closed orbits),
or $M_{i}$ is such that the restriction of $\psi_{\mathbb{R}}$ to $M_{i}$ is
minimal in the sense above (two such subsurfaces are visible in the example in
Figure 1 (b)). The latter are called _minimal components_ and there are at
most $g$ of them (where $g$ is the genus of $M$), see § 2.1.4.

The study of locally Hamiltonian flows is intertwined with the study another
famous class of flows on surfaces, namely _translation_ (_linear_) _flows_
333Translation flows are unit speed _linear flows_ on _translation surfaces_ ,
namely surfaces which are locally Euclidean outside a finite number of conical
singularities with cone angles of angle $2\pi k,k\in\mathbb{N}$. On these
surfaces, one has a well defined notion of direction and for each $\theta\in
S^{1}$ one can define a directional flow
$h^{\theta}_{\mathbb{R}}=(h_{t}^{\theta})_{t\in{\mathbb{R}}}$ which moves
points along lines in direction $\theta$ at unit speed. on translation
surfaces, which are at the center of Teichmüller dynamics. Each minimal
component of a locally Hamiltonian flow $\psi_{\mathbb{R}}$ indeed can be seen
as a _time-reparametrization_ (or a _time-change_) of a translation flow.
Notice though that the time-change is _singular_ at the fixed points
$\mathrm{Fix}(\psi_{\mathbb{R}})$ of $\psi_{\mathbb{R}}$ (see § 2.3.2 and
Remark 2.3 for a more precise description of the relation). One of the results
which can be inferred from classical results on translation flows (proved
through Teichmüller dynamics) is that the _typical_ flow (in the measure
theoretical sense) in $\mathscr{U}_{min}$ is _ergodic_ (with respect to $\mu$)
and the typical flow in $\mathscr{U}_{\neg min}$ is ergodic when restricted to
each minimal component (see § 2.1.4); it also follows that the associated
foliation into flow trajectories (or equivalently any Poincaré map of the
flow) is _uniquely_ ergodic (i.e. there is an unique invariant probability
transverse measure, the transverse measure induced by $\mu$). Notice, though,
that any locally Hamiltonian flow $\psi_{\mathbb{R}}$ with
$Fix(\psi_{\mathbb{R}})\neq\emptyset$ is _not_ uniquely ergodic (as a smooth
flow on a compact manifold): indeed, in the presence of singularities, there
are always trivial invariant measures (Dirac deltas) supported at
singularities. The presence of such measures and their effect on ergodic
integrals plays a key role in this work.

###  1.2. Power deviations and asymptotic behaviour of ergodic averages.

Let $\psi_{\mathbb{R}}$ denote either a locally Hamiltonian flow on $M$ in
$\mathscr{U}_{min}$ or the restriction of $\psi_{\mathbb{R}}$ in
$\mathscr{U}_{\neg min}$ to a minimal component $M_{i}$, that by abusing the
notation we will again denote by $M$ here, and assume that $\psi_{\mathbb{R}}$
is ergodic (and the associated foliation is uniquely ergodic). Thus, for every
smooth observable $f:M\to{\mathbb{R}}$ and for _almost every_
444Equidistribution of almost very point follows simply by ergodicity and
Birkhoff ergodic theorem. _Unique_ ergodicity yields a stronger conclusion
_if_ the observable if supported _outside_ the set of fixed points
$Fix(\psi_{\mathbb{R}})$: in this case equidistribution, namely (1.1), holds
for any _regular_ $p$, i.e. any $p$ such that its forward orbit is
$(\psi_{t}(p))_{t\geq 0}$ is dense). One can show though, that this is not the
case for observables $f$ which are non-zero at some fixed points, namely there
are regular points for which equidistribution does not hold.  initial point
$p\in M$, the ergodic averages of $f$ converge to the spatial averages, i.e.

(1.1) $\lim_{T\to+\infty}\frac{I_{T}(f,p)}{T}=\int_{M}f\mathrm{d}\,\mu,\
\text{where}\
I_{T}(f,p)=I_{T}(f,p,\psi_{\mathbb{R}}):=\int_{0}^{T}f(\psi_{t}(p))\mathrm{d}\,t.$

With _deviations of ergodic averages_ one refers to the study of the
_oscillations_ of the ergodic integrals $I_{T}(f,p)$ (or the related Birkhoff
sum over an interval exchange map obtained as Poincaré section) of an
observable $f:M\to\mathbb{R}$ of zero mean $\int_{M}f(p)\,\mathrm{d}\mu=0$
over the orbit of (typical) point $p\in M$. A distinctive phenomenon first
discovered experimentally by A. Zorich in the $1990$s (see [70] and also [40,
70]) is that deviations of ergodic averages have _polynomial nature_ , in the
following sense: for a typical flow, for suitable classes of observables, one
can find an exponent $\nu=\nu(f)$ with $0<\nu<1$ such that, $I_{T}(f,p)\sim
O(T^{\nu})$ for every regular point $p$, where we use the notation

(1.2) $I_{T}(f,p)\sim
O(T^{\nu})\qquad\Leftrightarrow\qquad\limsup_{T\to\infty}\frac{\log
I_{T}(f,p)}{\log T}=\nu.$

Kontsevich and Zorich explained this phenomenon heuristically using
renormalization and conjectured that, at least in the case of locally
Hamiltonian flows with non-degenerate fixed points555This is the framework
proposed in the paper [40], where Kontsevich (based on joint work with Zorich)
formulates the conjecture on the existence of the deviation spectrum (which
later became known as _Kontsevich-Zorich conjecture_). They first state the
result for homology classes (or equivalently characteristic functions over
interval exchange transformations) and then suggest that the phenomenon should
hold more generally if one considers, for simplicity, locally Hamiltonian
flows with Morse saddles and the space of smooth functions., there is a full
_deviation spectrum_ , namely there are exactly $g$ positive exponents
$0<\nu_{g}<\cdots<\nu_{2}<\nu_{1}:=1$ and a corresponding filtration of
$H_{g+1}\subset H_{g}\subset\cdots\subset H_{1}$ of the space of smooth
functions such that if $f\in H_{i}\backslash H_{i+1}$, with $1\leq i\leq g$,
then $I_{T}(f,p)\sim O(T^{\nu_{i}})$ (see [40]). Zorich gave in [71] a
rigorous proof of this phenomenon for ergodic integrals of a special class of
functions $f:M\to{\mathbb{R}}$, those which represent cohomology classes666In
the setting of [71], this class of functions reduces to the study of Birkhoff
sums of piecewise constant functions over interval exchange maps.. Forni
proved most of this conjecture in [23] (with the exception of _simplicity_ ,
namely the strict inequalities between $\nu_{g}<\nu_{g-1}<\dots<\nu_{1}$,
which was later proved by Avila and Viana in [5], while the positivity of
$\nu_{g}>0$ is a crucial part of [23]) for smooth observables and typical
flows in the closely related class of translation flows on translation
surfaces (see footnote 3). In the setting of locally Hamiltonian flows, he
considers the minimal case $\psi_{\mathbb{R}}\in\mathcal{U}_{min}$ and has the
further assumption that the (smooth) observable $f$ is _compactly supported_
outside of a neighbourhood of the finite set of fixed points
$\mathrm{Fix}(\psi_{\mathbb{R}})$ (or, more generally, in the Sobolev
regularity setting, that at least the function $f$ vanishes on
$\mathrm{Fix}(\psi_{\mathbb{R}})$, see [23] as well as [25]). We comment below
on the consequences of this assumption (see Remark 1.1).

The _power spectrum_ of ergodic integrals is related in [71, 23] to _Lyapunov
exponents_ of the Kontsevich-Zorich cocycle (so that in particular the strict
inequalities $\nu_{g}<\nu_{g-1}<\cdots<\nu_{1}$ hold in view of the simplicity
of the Lyapunov spectrum, which is the result later shown by Avila-Viana in
[5] work); the filtration is described by Forni in [23] in terms of kernels of
what we nowadays call _Forni’s invariant distributions_. We refer the
interested reader to [72, 24, 25, 5] for surveys of this phenomenon; in [24]
other instances of _parabolic_ flows for which deviations can be studied via
_renomalization_ are also mentioned.

A finer analysis of the behaviour of Birkhoff sums or integrals, beyond the
_size_ of oscillations, appears in the work [6] by Bufetov, as well as in the
work [45] by Marmi, Moussa and Yoccoz. In [6], Bufetov studies limit theorems
for ergodic integrals of translation flows (and describe weak limit
distributions) in terms of objects that he calls _Hölder cocycles_ (or, in the
more general context of Markov compacta, _finitely-additive measures_) and
turn out to be _dual_ to Forni’s invariant distributions (see [6] for
details). In particular, he shows that for a full measure set of translation
flows $h_{\mathbb{R}}:=(h_{t})_{t\in\mathbb{R}}$ (with respect to the Masur-
Veech measure), there exists $g-1$ cocycles777Here $\Phi_{i}(t,x)$ is a
_cocycle_ over the flow $h_{\mathbb{R}}$ in the sense that
$\Phi_{i}(t+s,x)=\Phi_{i}(t,x)+\Phi_{i}(s,h_{t}(x))$
$\Phi_{i}(t,x):{\mathbb{R}}\times M\to{\mathbb{R}}$, for $i=2,\dots,g$
(closely related to the _limit shapes_ introduced independently at the same
time by Marmi, Moussa and Yoccoz in [45]), each of which has a _pure power_
growth, i.e. such that $|\Phi_{i}(T,x)|\sim O(T^{\nu_{i}})$ (in the sense of
(1.2) above), which, together with the trivial cocycle $\Phi_{1}(t,x)=t$,
encode the _asymptotic behaviour_ of the ergodic integrals along the flow, by
providing an _asymptotic expansion_ up to subpolynomial terms, i.e. such that

$I_{T}(f,p,h_{\mathbb{R}})=\int_{0}^{T}f(h_{s}(p))\mathrm{d}s=c_{1}T+c_{2}\Phi_{2}(T,p)+\dots+c_{g}\Phi_{g}(T,p)+err(f,T,p),$

where the _error term_ $err(f,T,p)$ is subpolynomial, i.e. for any
$\epsilon>0$ there exists $C_{\epsilon}>0$ such that $|err(f,T,p)|\leq
C_{\epsilon}T^{\epsilon}$. The constant of the linear leading term is
$c_{1}=\int f\mathrm{d}\omega$, where $\omega$ is the underlying translation
surface area form, and the other coefficients can be computed evaluating
invariant distributions $D_{i}$ for $i=1,\dots,g$, i.e. $c_{i}=D_{i}(f)$.

###  1.3. Ergodicity of extensions.

A classical way to visualize and study the behaviour of ergodic averages of an
observable $f:M\to{\mathbb{R}}$ along the flow $\psi_{\mathbb{R}}$ on $M$ is
to consider the flow on $M\times{\mathbb{R}}$ given by coupling
$\psi_{\mathbb{R}}$ with the differential equation on ${\mathbb{R}}$

$\frac{\mathrm{d}\,\\!y}{\mathrm{d}\,\\!t}=f(\psi_{t}(p)),\qquad
y\in{\mathbb{R}},\ \quad t\in{\mathbb{R}}.$

One can see that the solution is given by the flow
$\Phi^{f}_{\mathbb{R}}:=(\Phi^{f}_{t})_{t\in\mathbb{R}}$ on
$M\times{\mathbb{R}}$ given by the formula

(1.3) $\Phi^{f}_{t}(p,y)=\left(\psi_{t}(p),\
y+\int_{0}^{t}f(\psi_{s}(p))\,\mathrm{d}s\right),\qquad p\in M,\
y\in{\mathbb{R}},\ t\in{\mathbb{R}}.$

Thus, the flow $\Phi^{f}_{\mathbb{R}}$ is a _skew product_ and provides an
_extension_ to $M\times{\mathbb{R}}$ of the flow $\psi_{\mathbb{R}}$ on $M$
(i.e. it _projects_ on the $M$ coordinate to the flow $\psi_{\mathbb{R}}$).
The motion in the ${\mathbb{R}}$ fiber is determined by the oscillations of
the _ergodic integrals_ of $f$ along $\psi_{\mathbb{R}}$. Notice that
$\Phi^{f}_{\mathbb{R}}$ preserves the _infinite_ product measure $\mu\times
Leb$, where $\mu$ is the invariant measure for $\psi_{\mathbb{R}}$ and $Leb$
denotes the Lebesgue measure on ${{\mathbb{R}}}$.

The study of these type of skew products goes back to Poincaré [52] and his
work on differential equations on $\mathbb{R}^{3}$ (in the case when
$\psi_{\mathbb{R}}$ is a smooth flow on the torus); the study of infinite skew
product extensions in greater generality became later a central topic in
_infinite ergodic theory_ , see for example the monographs [1, 59]. A basic
question is whether the flow $\Phi^{f}_{\mathbb{R}}$ is ergodic (see § 2.1.3)
or, if not, what is a description of ergodic components. A necessary condition
for ergodicity is that $f$ has zero mean, i.e. $\int_{M}f\mathrm{d}\mu=0$,
since otherwise $\Phi^{f}_{\mathbb{R}}$ has a _drift_ and is not even
_recurrent_ (see § 2.1.3). In the setting of extensions, a property completely
opposite to ergodicity is _reducibility_. If the skew product on
$M\times{\mathbb{R}}$ is _reducible_ (see § 2.1.3 for the definition),
$M\times{\mathbb{R}}$ is foliated into invariant sets for
$\Phi^{f}_{\mathbb{R}}$, on which the dynamics is conjugated to
$\psi_{\mathbb{R}}$ on $M$.

Taking a suitably chosen Poincaré section (see § 2.3.3 for details), the
ergodicity of $\Phi^{f}_{\mathbb{R}}$ is equivalent to the ergodicity of a
skew product automorphism $T_{\varphi}$ of the strip $I\times{\mathbb{R}}$,
where $I=[0,1)$, of the form

(1.4) $T_{\varphi}(x,y)=(T(x),y+\varphi(x)),\qquad x\in I,\quad
y\in{\mathbb{R}},$

where $T:I\to I$ is a _rotation_ (i.e. the map $T(x)=x+\alpha\mod 1$) when $M$
is a torus ($g=1$), or more in general, for any $g\geq 1$, an _interval
exchange transformation_ (see § 2.2.1), while
$\varphi:I\to\overline{{\mathbb{R}}}$ is a function with _singularities_ (i.e.
points where the function blows up) which are are _logarithmic_ (see § 2.3.1
for the precise definition) whenever $\psi_{\mathbb{R}}$ has only non-
degenerate saddles (while _polynomial_ in presence of a degenerate saddle).

######  Remark 1.1.

Notice also that if $f$ is compactly supported in
$M\backslash\mathrm{Fix}(\psi_{\mathbb{R}})$ (or, more generally, it vanishes
on $\mathrm{Fix}(\psi_{\mathbb{R}})$, see § 4.2.2, in particular Proposition
4.1, then the function $\varphi$ in (1.4) is piecewise absolutely continuous
(or even piecewise smooth), in particular does not have logarithmic
singularities. Thus, the singularities are a combined effect of the nature of
the locally Hamiltonian parametrization, _together with_ the assumption that
(the jet of) $f$ does not vanish identically zero at
$\mathrm{Fix}(\psi_{\mathbb{R}})$.

We stress that the problem of ergodicity of skew product extensions over IETs
is currently actively researched, but still widely open. See for example [29,
10, 21, 17, 28, 53, 54, 7] for some results in particular settings.

In the _genus one_ case, the existence of ergodic skew products was first
discovered by Krygin, in [42], in the case where the flow $\psi_{\mathbb{R}}$
has no singularities. Ergodicity of extensions of typical _Arnold flows_
888Recall that an Arnold flow is the restriction to the minimal component of a
locally Hamiltonian flow in genus one with one saddle and one center, see
Figure 1(a). (or, correspondingly, of skew products of the form (1.4) where
$T$ is a rotation and $\varphi$ has one _asymmetric_ logarithmic singularity,
see § 2.3.1 for definitions), was proved by Fayad and Lemańczyk in [14], where
they proved ergodicity for a full measure set of rotation numbers. This case
is particularly delicate since the underlying Arnold flows are mixing; in a
related easier case (namely the case when $T$ is a rotation but $\varphi$ in
(1.4) has one _symmetric_ logarithmic singularity, see § 2.3.1), ergodicity
was proved previously by Lemańczyk and the first author, see [19].

Very little is understood in the case of infinite skew product extensions
(i.e. extensions by a non-compact fiber, for which the natural invariant
measure is infinite) of locally Hamiltonian flows in higher genus $g\geq 2$,
even in the case when $f:M\to{\mathbb{R}}$ has compact support in
$M\backslash\mathrm{Fix}(\psi_{\mathbb{R}})$ and the cocycle $\varphi$ is
piecewise-smooth (see Remark 1.1) or even piecewise-constant. Some specific
results for piecewise constant or piecewise absolutely continuous cocycles
over IETs with $d>2$ were proved for example in [10, 21, 22, 44].

We considered the case of a locally Hamiltonian flow $\psi_{\mathbb{R}}$ with
non-degenerate saddles and a general observable $f:M\to{\mathbb{R}}$ and,
correspondingly, of a cocycle $\varphi$ with _logarithmic_ (symmetric)
_singularities_ in our previous joint work [20], where we showed the
_existence_ of ergodic extensions in any genus, but for a very restrictive
class of locally Hamiltonian flows. More precisely, in [20] we could treat
only the special (measure zero) class of locally Hamiltonian flows in
$\mathscr{U}_{min}$ for which the Poincaré section can be chosen to be a
_self-similar_ interval exchange transformation999These IETs are also known as
_periodic-type_ IETs in the literature, see for example [60]. In [20] we
further assume that the periodic-type IET is of _hyperbolic type_ , see [20]
for details. Explicit examples of locally Hamiltonian flow of hyperbolic
periodic type were constructed in [10]. and restrict the observable $f$ to
belong to an infinite dimensional (but finite codimension $g$) space. For
extensions of flows in this special class, though, we could provide a complete
description of the ergodic behavior and prove a dichotomy between ergodicity
and reducibility. One of the main results of this paper is to show that this
dichotomy actually holds also for a _full measure_ set of such minimal locally
Hamiltonian flows (see the Main Theorem 1.2 below).

###  1.4. Main results

One of the main results of this paper is that infinite _ergodic_ extensions
_exist_ in any genus $g\geq 1$ for a _full measure_ set of (minimal) locally
Hamiltonian flows with non-degenerate fixed points (with respect to the Katok
fundamental class for each _stratum_ , see § 2.1.1). More precisely, we are
able to extend the result previously proved in [20] only for a _measure zero
class_ of self-similar IETs to a _full measure_ set of locally Hamiltonian
flows, by proving the following _dichotomy_ for the dynamics of the
extensions:

######  Theorem 1.2 (Ergodic or reducible extensions of locally Hamiltonian
flows).

For a _full measure_ set of locally Hamiltonian flows $\psi_{\mathbb{R}}$ with
non-degenerate saddles in $\mathcal{U}_{min}$, for any $\epsilon>0$, for any
$f$ in a infinite dimensional (finite codimension) subspace
$K\subset\mathscr{C}^{2+\epsilon}(M)$, we have the following dichotomy:

  * •

If $\sum_{x\in\mathrm{Fix}(\psi_{\mathbb{R}})}|f(x)|\neq 0$ then the extension
$\Phi^{f}_{{\mathbb{R}}}$ is ergodic;

  * •

If $\sum_{x\in\mathrm{Fix}(\psi_{\mathbb{R}})}|f(x)|=0$ then the extension
$\Phi^{f}_{{\mathbb{R}}}$ is reducible.

We will comment later on the full measure set, which is explicitly described
by a new _Diophantine-type_ condition (see § 3.2.2 for the definition) as well
as on the infinite dimensional (invariant) subspace $K$ (which will be defined
as the kernel of $g$ invariant distributions, see § 7.2).

The proof of this ergodicity result takes as starting point our results on
deviations of ergodic averages101010In particular, to prove ergodicity we need
to show a form of _tightness_ of Birkhoff sums, which, combined with enough
_oscillations_ thanks to the presence of logarithmic singularities, allows to
apply classical _essential values_ (see [59]). of $f$, which is of independent
interest and we now state. As it is clear from the dichotomy, to produce
ergodic extensions one needs to study observables $f:M\to{\mathbb{R}}$ which
_do not vanish_ at (at least one) the saddle points111111Since we are here
assuming that $\psi_{\mathbb{R}}\in\mathcal{U}_{min}$ has only non-degenerate
fixed points, $\mathrm{Fix}(\psi_{\mathbb{R}})$ consists of simple saddles
only, see § 2.1.1. in $\mathrm{Fix}(\psi_{\mathbb{R}})$.

For ergodic integrals of (typical) minimal locally Hamiltonian flows in
$\mathcal{U}_{min}$ (see Theorem 1.3), as well as for minimal components of
(typical) locally Hamiltonian flows in $\mathcal{U}_{\neg min}$ (see Theorem
1.4), we give asymptotic descriptions of the deviation spectrum, as follows.

######  Theorem 1.3 (Asymptotic power spectrum of ergodic integrals (minimal
case)).

For a full measure set of locally Hamiltonian flows on $M$ in
$\mathscr{U}_{min}$ with non-degenerate saddles, there exist a _power
spectrum_ $0<\nu_{g}<\cdots<\nu_{2}<\nu_{1}:=1$, where $g$ is the genus of the
surface $M$ and, for any $\epsilon>0$, _invariant distributions_
$D_{i}:C^{2+\epsilon}(M)\to{\mathbb{R}}$, $i=1,\ldots,g$, such that, for every
$f\in C^{2+\epsilon}(M)$, we have the _asymptotic expansion_ :

(1.5)
$\int_{0}^{T}f(\psi_{t}(x))\,dt=\sum_{i=1}^{g}D_{i}(f)u_{i}(T,x)+\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}f(\sigma)u_{\sigma}(T,x)+err_{b}(f,T,x),$

where, for $1\leq i\leq g$, $u_{i}$ are smooth cocycles
$u_{i}:{\mathbb{R}}\times M\to{\mathbb{R}}$ over the flow $\psi_{\mathbb{R}}$
such that

(1.6)
$\limsup_{T\to+\infty}\frac{\log\|u_{i}(T,\,\cdot\,)\|_{L^{\infty}(M)}}{\log
T}={\nu_{i}},$

while, for $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$, $u_{\sigma}$ are smooth
cocycles $u_{\sigma}:{\mathbb{R}}\times M\to{\mathbb{R}}$ over
$\psi_{\mathbb{R}}$ which grow sub-polynomially pointwise and in $L^{p}$ norm
for every $p\geq 1$, i.e. such that

(1.7) $\limsup_{T\to+\infty}\frac{\log|u_{\sigma}(T,x)|}{\log T}=0\text{ for
a.e. }x\in
M,\quad\text{and}\quad\limsup_{T\to+\infty}\frac{\log\|u_{\sigma}(T,\,\cdot\,)\|_{L^{p}(M)}}{\log
T}=0,\text{for all}\ p\geq 1,$

and $err_{b}$ is a uniformely bounded _error_ term, i.e.

(1.8)
$\sup_{t\in{\mathbb{R}}}\|err_{b}(f,t,\,\cdot\,)\|_{L^{\infty}}<+\infty.$

Furthermore, for every $\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$ and for
$\mu$-almost every $x\in M$, the values of the cocyle $t\mapsto
u_{\sigma}(t,x)$ are _equidistributed_ on ${\mathbb{R}}$, i.e. for any pair of
intervals $J_{1},J_{2}\subset{\mathbb{R}}$ we have

(1.9) $\lim_{T\to+\infty}\frac{Leb\\{t\in[0,T]:u_{\sigma}(t,x)\in
J_{1}\\}}{Leb\\{t\in[0,T]:u_{\sigma}(t,x)\in
J_{2}\\}}=\frac{|J_{1}|}{|J_{2}|}.$

Finally, if we set

(1.10)
$err(f,t,x):=\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})}f(\sigma)u_{\sigma}(t,x)+err_{b}(f,t,x),$

as soon as $f$ does not vanish identically on
$\mathrm{Fix}(\psi_{\mathbb{R}})$, for $\mu$-almost every $x\in M$ also the
values of the cocyle $t\mapsto err(f,t,x)$ are _equidistributed_ on
${\mathbb{R}}$.

Main Theorem 1.3 completes in particular the proof of the Kontsevich-Zorich
conjecture, in its original formulation for smooth functions over locally
Hamiltonian flows with non-degenerate saddles (as formulated in [40], see the
above § 1.2). The result should be seen as a generalization (for
smooth121212The class of functions considered by Forni [23, 25] and Bufetov
[6] is more general: smoothness is not required, but only a Sobolev condition
in [23] (see also [25] for a more general result on the cohomological
equation) and a _weak Lipschitz property_ in Bufetov’s work, see [6] for
details. functions) of both the results by Forni [23] (since it proves the
existence of a power deviation spectrum) and Bufetov [6] (since we show the
existence of asymptotic cocycles). While the observables in both Forni’s [23]
and Bufetov’s [6] works vanish on $\mathrm{Fix}(\psi_{\mathbb{R}})$, we allow
the observables to be non-zero at singularities in
$\mathrm{Fix}(\psi_{\mathbb{R}})$. This leads to the presence in the
asymptotic expansion of $k$ new cocycles, where $k$ is the cardinality of
$\mathrm{Fix}(\psi_{\mathbb{R}})$, one for each saddle
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$. We will call these $u_{\sigma}$
_singular cocycles_ , since they describe the fluctuations of the ergodic
averages due to the presence of singularities. While these cocycles
$u_{\sigma}$ have sub-polynomial deviations, as shown by (1.7), they are _not_
uniformly bounded.

Comparison to Forni’s and Bufetov’s works. To further compare the result with
Forni’s [23] and Bufetov’s [6] works, let us consider the global error term
$err(f,t,\cdot)$ defined as in (1.10) combining the bounded error
$err_{b}(f,t,\cdot)$ together with the cocycles $u_{\sigma}$,
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$. Then one can see that
$err(f,t,\cdot)$ has always _sub-polynomial_ pointwise growth (in view of
(1.7) combined with (1.8)), but we have a _dichotomy_ : on one hand, if $f$
does vanish identically on $\mathrm{Fix}(\psi_{\mathbb{R}})$, $err(f,t,\cdot)$
coincides with $err_{b}(f,t,\cdot)$ and is uniformly bounded. In this case,
the $g$ cocycles $u_{i}$, which lead the power growth, can be shown _a
posteriori_ to coincide with the _Bufetov functionals_ in [6] up to a bounded
error. On the other hand, as soon as $f$ does _not_ vanish identically on
$\mathrm{Fix}(\psi_{\mathbb{R}})$, $err(f,t,x)$ _cannot_ be controlled
_uniformly_ : for $\mu$-almost every $x$, the function $t\mapsto err(f,t,x)$
is unbounded, in view of the equidistribution of $err(f,t,\cdot)$ in this case
(see the final part of Theorem 1.3, which follows directly from the ergodicity
of the extensions proved in the Main Theorem 1.2, more precisely from an
application of the _ratio_ ergodic theorem in infinite ergodic theory).

This novel phenomenon is an effect of the presence of infinite _tails_ , due
to the assumption that $f$ is non-zero at (some) singularities and the slowing
down of trajectories near Hamiltonian saddles. We are nevertheless able to
control the error term $err(f,t,\cdot)$ _pointwise_ almost everywhere (in view
of (1.12)) and _in average_ , in any $L^{p}$ norm with $p\geq 1$, in view of
(1.13).

Minimal components in the non-minimal setting. Another novelty of our work is
that, while Forni and Bufetov in [23, 6] study only minimal flows, we prove
the existence of an asymptotic expansion also for ergodic integrals of non-
mimimal flows in $\mathcal{U}_{\neg min}$. More precisely, we prove the
following result for a minimal component $M_{0}\subset M$ of a _typical_ flow
on $\mathcal{U}_{\neg min}$.

######  Theorem 1.4 (Asymptotic power spectrum for non minimal components).

For a full measure set of locally Hamiltonian flows on $M$ in
$\mathscr{U}_{\neg min}$ with non-degenerate saddles, for any minimal
component $M_{0}\subset M$ of $\psi_{\mathbb{R}}$, if $g_{0}$ denotes the
genus of $M_{0}$, there exist a _power spectrum_
$0<\nu_{g_{0}}<\cdots<\nu_{2}<\nu_{1}:=1$ and, for any $\epsilon>0$, $g_{0}$
_invariant distributions_ $D_{i}:C^{2+\epsilon}(M_{0})\to{\mathbb{R}}$,
$i=1,\ldots,g_{0}$, and $g_{0}$ smooth cocycles $u_{i}:{\mathbb{R}}\times
M_{0}\to{\mathbb{R}}$, for $i=1,\ldots,g_{0}$, each of which satisfies (1.6),
such that for every $f\in C^{2+\epsilon}(M_{0})$ we have an asymptotic
expansion

$\int_{0}^{T}f(\psi_{t}(x))\,dt=\sum_{i=1}^{g_{0}}D_{i}(f)u_{i}(T,x)+err(f,T,x),$

where, if $f$ vanishes on $\mathrm{Fix}(\psi_{\mathbb{R}})\cap M_{0}$, the
error term $err(f,T,\cdot)$ satisfies

(1.11)
$\limsup_{T\to+\infty}\frac{\log\|err(f,T,\,\cdot\,)\|_{L^{\infty}(M_{0})}}{\log
T}\leq 0,$

while if $f$ is not identically zero on $\mathrm{Fix}(\psi_{\mathbb{R}})\cap
M_{0}$ then

(1.12) $\limsup_{T\to+\infty}\frac{\log|err(f,T,x)|}{\log T}=0\text{ for
$\mu$-almost every }x\in M_{0},$

and furthermore

(1.13)
$\limsup_{T\to+\infty}\frac{\log\|err(f,T,\,\cdot\,)\|_{L^{p}(M_{0})}}{\log
T}=0\text{ for every }p\geq 1.$

Notice that in this case, when restricting to a minimal component of
$\psi_{\mathbb{R}}\in\mathcal{U}_{\neg min}$, we only claim that
$err(f,T,\cdot)$ grows sub-polynomially (which is the same type of estimate
proved by Bufetov for the error term in the symmetric case). This result is in
particular an extension of Bufetov’s work [6] to the restriction to a minimal
component in the non minimal case $\psi_{\mathbb{R}}\in\mathcal{U}_{\neg
min}$.

Thus, Theorems 1.3 and 1.4 complete the study of deviations of ergodic
averages of smooth functions over locally Hamiltonian flows with _non-
degenerate_ saddles. The study of locally Hamiltonian flows with _degenerate-
saddles_ leads to other new phenomena and additional polynomial terms in the
asymptotic expansion and is treated in an upcoming paper by M. Kim and the
first author [18].

On the proof and the Diophantine-like conditions. The proof of the asymptotic
expansion in Theorem 1.3, which will be proved at the same time than Theorem
1.4, follows a completely different approach to both Forni’s [23] and
Bufetov’s [6] works and is inspired by Marmi-Moussa-Yoccoz work [44] on
solving the cohomological equation for (Roth-type) interval exchange
transformations (and the follow up work [48] by Marmi and Yoccoz). We comment
in detail on this strategy below in § 1.5.

An advantage of this different approach is that it allows to give a
_description_ of the full measure set of locally Hamiltonian flows for which
the result holds in terms of a _Diophantine-type_ condition. Furthermore, it
also provides a different construction of the cocycles which describe the
asymptotic behaviour of ergodic integrals in terms of the correction
operators.

The full measure _Diophantine-like_ conditions (which are different for
Theorem 1.3 and Theorem 1.4 respectively) are expressed more precisely on the
interval exchange transformations which arise as Poincaré sections of the
flows. We introduce (in § 3.2) two such conditions, both of which we show to
be of full measure. The first, that we call _Uniform Diophantine Condition_
(or LABEL:UDC), is used to prove the existence of the asymptotic expansion in
both Theorem 1.3 and Theorem 1.4 up to a subpolynomial error. In the case of
minimal flows in $\mathcal{U}_{min}$, to improve the estimates on the error
and show in particular that the error is equidistributed (see the second part
of Theorem 1.3), we need to assume a more restrictive condition, namely the
_Symmetric Uniform Diophantine Condition_ (or LABEL:SUDC). For this result
indeed we also need to crucially exploit the cancellations proved by the
second author in [63] to prove typical absence of mixing and these require
further assumptions on the IET to hold.

Both Diophantine-like conditions expressed in terms of the matrices of the
Rauzy-Veech cocycle, which often plays the role of _multi-dimensional
continued fraction_ in the study of IETs. These conditions, similarly to the
Roth-type condition for IETs introduced by Marmi-Moussa and Yoccoz in [44]
(and its variations, see for example [44, 46, 48, 47]), impose constraints
both on the growth of the matrices of (an acceleration of) the cocycle, as
well as requests on the hyperbolic behaviour of the matrix product, in the
form of Oseledets genericity requests. In addition, we require _effective_
Oseledets control, which in turns allow to control certain _Diophantine
series_ (see § 3.3). We point out that similar conditions also appear in the
recent work [27] on rigidity of generalized interval exchanges.

###  1.5. Correction of cocycles with logarithmic singularities

We comment now on the methods and the proofs. First of all we work with
Poincaré maps, both to study the flow $\psi_{\mathbb{R}}$ and its extensions
$\Phi^{f}_{\mathbb{R}}$; it is well known that Poincaré maps of area-
preserving flows, in suitably chosen coordinates, are _interval exchange
transformations_ (for short IETs), namely, piecewise-isometries of the
interval $I=[0,1)$ (the definition is recalled in § 2.2.1). Moreover, any
minimal locally Hamiltonian flow admits a representation as _special flow_
over the IET $T:I\to I$ which arise as Poincaré map (see § 2.2.2 for
definitions). The _roof function_ $r:I\to\overline{{\mathbb{R}}^{+}}$ which
arise from this representation has _singularities_ at the discontinuities of
$T$, which, in case of simple (non-degenerate) saddles, are of _logarithmic
type_ (formally defined in § 2.3.1), i.e. as $x\to x_{i}^{\pm}$ approaches a
discontinuity $x_{i}\in I$ of $T$ from the right or left, $r(x)$ blows up as
$C_{i}^{\pm}|\log(x-x_{i})|$ , where the constants $C_{i}^{\pm}$ are
_positive_ and are globally _symmetric_ , namely
$\sum{C}_{i}^{+}=\sum{C}_{i}^{-}$, for typical flows in $\mathcal{U}_{min}$,
while _asymmetric_ for minimal components of typical flows in
$\mathcal{U}_{\neg min}$.

Fix now an observable $f:M\to{\mathbb{R}}$ which is non-zero on
$\mathrm{Fix}(\psi_{\mathbb{R}})$. To study ergodic integrals, we build the
extension $\Phi^{f}_{\mathbb{R}}$ on $M\times{\mathbb{R}}$ (given by (1.3)).
Choosing a Poincaré section for the extension which projects on $I$, namely of
the form $I\times{\mathbb{R}}$, the Poincaré first return map of
$\Phi^{f}_{\mathbb{R}}$ (in suitable coordinates) turns out to be a _skew
product_ over the IET $T$ of the form (1.4), in which the _cocycle_ $\varphi$
has _logarithmic singularities_ (where the constants $C_{i}^{\pm}$ here can be
positive or negative, or zero if the function is zero on
$\mathrm{Fix}(\psi_{\mathbb{R}})$, in which case there are no singularities,
see Remark 1.1). We have now reduced the study of ergodic integrals and
ergodicity of extensions to the study of Birkhoff sums of cocycles with
logarithmic singularities over IETs and ergodicity of skew products over IETs
with logarithmic singularities.

Under the new Diophantine-type conditions that we introduce in § 3.2, for
every function with logarithmic singularities, we prove the existence of a
_correction_ operator, namely an operator which, removing the projection on a
finite dimensional space (which corresponds morally to the projection on the
unstable space of renormalization), allows to get a better control of the
behaviour of Birkhoff sums of functions with logarithmic singularities (see
Theorem 6.1 for the precise statement). The result provides an extension of
the main result in the work of Marmi-Moussa-Yoccoz [44]. In the latter, in
order to solve the cohomological equation for IETs of Roth-type, correction
operators are constructed for (piecewise) _absolutely continuous_ cocycles.

While the main steps of our correction procedure are inspired by the
construction introduced in [44] (and later developed in [48]), there are
considerable differences and difficulties. Notably, while the authors of [44]
were interested in controlling the growth of the sequence of Birkhoff sums
$(S(k)\varphi)_{k\in\mathbb{N}}$ of a piecewise absolutely continuous
functions $\varphi$ using the _uniform norm_ (in order to keep them bounded
after correction and be able to apply Gottschalk-Hedlund theorem, see [44] for
details), for functions with _logarithmic singularities_ , the uniform norm
_cannot_ be used (since functions with logarithmic singularities are always
unbounded). The key idea to treat cocycles with logarithmic singularities in
this paper is to exploit instead the $L^{1}$-norm and to build correction
operators which allow to bound or control the $L^{1}$-norm of the sequence
$(S(k)\varphi)_{k\in\mathbb{N}}$. The use of the $L^{1}$-norm has already
appeared in our previous work [20], where we had considered the correction
problem131313In [20], for IETs of hyperbolic periodic type, we build
correction operators for cocycles with _symmetric_ logarithmic singularities
and we then exploit the result to build ergodic extensions, but we do not work
out the full deviation spectrum and asymptotic cocycles formalism. for the
(measure zero set of) IETs of hyperbolic periodic type. It turns out that to
extend the result to almost every IET requires once again changes in the basic
step of construction, as well as the introduction of the above mentioned
delicate Diophantine-type condition on the IET. We refer the interested reader
to § 6 (and in particular the outline of the strategy to build the correction
operators given in § 6.1.2) for further details on the differences and the
steps in the construction of the correction operators.

The construction of the asymptotic cocycles $u_{i}:{\mathbb{R}}\times
M\to{\mathbb{R}}$ which lead to understanding the behaviour of ergodic
integrals (see the statement of Main Theorem 1.3) is strictly connected to the
finite dimensional space of _corrections_. Indeed, corrections can be realized
by subtracting _piecewise constant cocycles_ , which, through the
correspondence between extensions and skew-products, allow to define the
asymptotic cocycles $u_{i}$.

In the case of minimal locally Hamiltonian flows in $\mathcal{U}_{min}$ (which
give rise to _symmetric_ logarithmic singularities), we also exploit the
delicate _cancellations_ among contributions of singularities which were
proved by the second author in [63] and, introducing the LABEL:SUDC
Diophantine-type condition, we are able to prove that, after corrections, a
subsequence of Birkhoff sums $(S(k)\varphi)_{k\in\mathbb{N}}$ is _tight_.
Tightness, combined with partial rigidity of the IET in the base (a result
which dates back to Katok [33]) and the _presence_ of logarithmic
singularities (which comes from the assumption that $f$ is non identically
zero on $\mathrm{Fix}(\psi_{\mathbb{R}})$), allows to apply a quite standard
ergodicity criterium based on the existence of _essential values_ (see
Proposition 8.4 for the precise incarnation of the criterium which we use in
this paper). This allows to prove ergodicity of the corresponding extensions.

### Structure of the paper

In § 2 we recall basic definitions and background material on locally
Hamiltonian flows and their extensions. We also summarize their typical
ergodic properties and explain the reductions to special flows and skew
products over IETs. In § 3, after recalling the required definitions and
properties of the Rauzy-Veech induction procedure and the associated cocycle,
we define the two Diophantine conditions (the LABEL:UDC and the LABEL:SUDC
conditions) and prove that they have full measure.

In §§ 4, 5 and 6 we study cocycles with logarithmic singularities over IETs.
After giving definitions and proving elementary properties in § 4, we proceed
in § 5 at investigating the renormalization process induced on such cocycles
by performing Rauzy-Veech induction. The correction operators are constructed
in § 6 (where the above mentioned Theorem 6.1 about existence and properties
of the correction operators is proved).

The asymptotic deviation spectrum (see the first part of Main Theorem 1.3) is
proved in § 7.2, where the asymptotic of ergodic integrals is recovered from
the cocycles associated to the correction operators. In § 8 we state the
ergodicity criterium that we then apply to prove ergodicity of extensions.
After discussing also the reducibility case, we then prove Main Theorem 1.2,
as well as the second part of Main Theorem 1.3. Some technical but standard
proofs in this part are relegated to the Appendix (in particular the proofs of
the ergodicity criterium and of a cohomological reduction result which is
needed for the reducibility part).

##  2\. Definitions, background material and reductions

In this section we recall some basic definitions and background material
concerning locally Hamiltonian flows (§ 2.1) and their extensions (§ 2.1.3),
including a brief summary in § 2.1.4 of our current knowledge of their typical
chaotic properties. We also the definition of special flows (see § 2.2.2) and
skew-products (in § 2.2.3) over interval exchange transformations (defined in
§ 2.2.1). We finally recall in § 2.3 the representation of locally Hamiltonian
flows to special flows (see § 2.3.2) with logarithmic singularities (defined
in § 2.3.1) and the reduction of the study of their extensions to skew
products over IETs, see § 2.3.3.

###  2.1. Locally Hamiltonian flows

Let $(M,\omega)$ be a surface with a fixed smooth area form $\omega$. A
_smooth area preserving flow_ $\psi_{\mathbb{R}}=(\psi_{t})_{t\in\mathbb{R}}$
on $M$ is a smooth flow on $M$ which preserves the measure $\mu$ associated to
$\omega$. These flows are also called _locally Hamiltonian flows_ or _multi-
valued Hamiltonian flows_ in the literature, in view of their interpretation
as flows locally given by Hamiltonian equations, see the introduction.

It turns out that such smooth area preserving flows on $M$ are in one-to-one
correspondence with smooth closed real-valued differential $1$-forms as
follows. Given a smooth, closed, real-valued differential $1$-form $\eta$, let
$X$ be the vector field determined by $\eta=i_{X}\omega$ where $i_{X}$ denotes
the contraction operator, i.e. $i_{X}\omega=\omega(\eta,\cdot)$ and consider
the flow $\psi_{\mathbb{R}}$ on $M$ given by $X$. Since $\eta$ is closed, the
transformations $\psi_{t}$, $t\in\mathbb{R}$, are area-preserving. Conversely,
every smooth area-preserving flow can be obtained in this way.

Let $\mathrm{Fix}(\psi_{\mathbb{R}})$ denote the set of _fixed points_ (also
called _singularities_) of the flow $\psi_{\mathbb{R}}$. We will always
require that $\mathrm{Fix}(\psi_{\mathbb{R}})$ is a _finite set_ , so in
particular singularities are _isolated_. Remark that when $g\geq 2$,
$\mathrm{Fix}(\psi_{\mathbb{R}})$ is always not empty, thus singularities are
isolated. Since $\psi_{\mathbb{R}}$ is area-preserving, _singularities_ in
$\mathrm{Fix}(\psi_{\mathbb{R}})$, as shown in Figure 2, can be either centers
(Fig. 2(a)), simple saddles (Fig. 2(b)) or multi-saddles (i.e. saddles with
$2k$ pronges, $k\geq 2$, see Fig. 2(c) for $k=3$). For $g=1$, i.e. on a torus,
if there is a singularity then there has to be another one and we get an
Arnold flow as in Figure 1(a).

(a) center

(b) simple saddle

(c) multisaddle

Figure 2. Type of singularities of a locally Hamiltonian flow.

We call _saddle connection_ a flow trajectory from a saddle to a saddle and a
_saddle loop_ a saddle connection from a saddle to the same saddle (see Fig.
3). A _periodic component_ is either a (maximal) punctured disk or a (maximal)
cylinder filled with closed (i.e. periodic) trajectories (see Fig. 3(a) and
Fig. 3(b) respectively). A _minimal component_ is a subsurface
$M^{\prime}\subset M$, possibly with boundary, such that any trajectory
different than a fixed point is _dense_ in $M^{\prime}$. Periodic and minimal
components are bounded by union of saddle connections.

(a) periodic island

(b) periodic cylinder

(c) $g=2$ minimal component

Figure 3. Periodic and minimal components.

####  2.1.1. Open sets, genericity and minimality

Let us denote by $\mathcal{F}$ the set of smooth closed $1$-forms on $M$ (i.e.
locally Hamiltonian flows) with isolated zeros. One can define a _topology_ on
$\mathcal{F}$ by considering perturbations of closed smooth $1$-forms by
(small) closed smooth $1$-forms141414Let $\eta$, $\eta^{\prime}$ be two smooth
closed $1$-forms. We say that $\eta^{\prime}$ is an $\epsilon$-perturbation of
$\eta$ if for any $x\in M$ there exists coordinates on a simply connected
neighbourhood $U$ of $x$, such that $\eta|_{U}=dH$ and
$(\eta^{\prime}-\eta)|_{U}=dh$ where
$\|h\|_{C^{\infty}}<\epsilon\|H\|_{C^{\infty}}$.. We say that a condition is
_generic_ (in the sense of Baire) if it holds for flows described by an open
and dense set of forms with respect to this topology.

Let $\mathcal{A}\subset\mathcal{F}$ be the subset of Morse $1$-forms (adopting
the notation introduced by Ravotti [56]), namely forms which are locally the
differential of a _Morse function_ (i.e. a function that has _non-degenerate_
zeros, so that the Hessian at every fixed point is non-degenerate). The set
$\mathcal{A}$ of Morse $1$-forms is then _generic_. Locally Hamiltonian flows
corresponding to forms in $\mathcal{A}$ have only _non-degenerate fixed
points_ , i.e.  _centers_ and _simple saddles_ (see Figures 2(a) and 2(b)), as
opposed to degenerate _multi-saddles_ (as in Fig. 2(c)). We denote by
$\mathcal{A}_{s,c}$ the set of $1$-forms in $\mathcal{A}$ with $s$ saddle
points and $c$ centers. By the Poincare-Hopf Theorem, $c-s=2-2g$. Furthermore,
each $\mathcal{A}_{s,l}$ is open and their union $\mathcal{A}$ is dense in
$\mathcal{F}$ (see e.g. Lemma 2.3 in [56]).

For every $1$-form in $\mathcal{A}$, the surface $M$ splits into periodic
components and (up to $g$) minimal components (as proved independently by
Maier [43], Levitt [41] and Zorich [72]). Notice that if there is a unique
minimal component (which is equal to the whole surface $M$), then $c=0$ (since
if there is a center is associated to a periodic component) and $s=2g-2$.

Moreover, one can show that if the flow $\psi_{\mathbb{R}}$ given by a closed
$1$-form $\eta$ has a _saddle loop homologous to zero_ (i.e. the saddle loop
is a _separating_ curve on the surface), then the saddle loop is persistent
under small perturbations (see § 2.1 in [72] or Lemma 2.4 in [56]). In
particular, the set of locally Hamiltonian flows which have at least one
saddle loop is an open set, which consists of _non-minimal_ flows. The set
$\mathscr{U}_{\neg min}$ mentioned in the introduction is an open and dense
set of this open set (where the open condition guarantees _asymmetry_ in the
special flow representation recalled in § 2.3.2, we refer to [56] for the
precise definition, see Notation 3.3 in § 3.1 of [56]). The set
$\mathscr{U}_{min}$ is given by the interior (which one can show to be non-
empty) of the complement of $\mathscr{U}_{\neg min}$, i.e. the set of locally
Hamiltonian flows without saddle loops homologous to zero151515Note that
saddle loops _non-homologous_ to zero (as well as saddle connections)
disappear after arbitrarily small perturbations; therefore neither the set of
1-forms with saddle loops (or more generally saddle connections) non-
homologous to zero, nor its complement are open (see [56] for details)..

####  2.1.2. Measure class and typicality

Let us fix an open set $\mathcal{A}_{s,c}$ of closed $1$-forms with $c$
centers and $s$ (simple) saddles. A _measure-theoretical notion of typical_ on
$\mathcal{A}_{s,c}$ can be defined on each $\mathcal{A}_{s,c}$ as follows, by
using the _Katok fundamental class_ (introduced by Katok in [32], see also
[50]), i.e. the cohomology class of the 1-form $\eta$ which defines the flow.
Let $\gamma_{1},\dots,\gamma_{n}$ be a base of the relative homology
$H_{1}(M,\mathrm{Fix}(\psi_{\mathbb{R}}),\mathbb{R})$, where $n=2g+s+c-1$, and
consider the period map

$\Theta(\eta)=\Big{(}\int_{\gamma_{1}}\eta,\ldots,\int_{\gamma_{n}}\eta\Big{)}\in{\mathbb{R}}^{n}.$

The map $\Theta$ is well defined in a neighbourhood of $\eta$ in
$\mathcal{A}_{s,c}$ and one can show that it is a complete isotopy invariant
(see [32], or also Prop. 2.7 in [56]).

The pull-back $Per_{*}Leb$ of the Lebesgue measure class (i.e. class of sets
with zero measure) by the period map gives the desired measure class on closed
$1$-forms in $\mathcal{A}_{s,c}$. When we use the expression _typical_ below
(or typical in $\mathcal{U}_{min}$ or $\mathcal{U}_{\neg min}$) we mean full
measure in each $\mathcal{A}_{s,c}$ with respect to this measure class on each
$\mathcal{A}_{s,c}$ (or on each open subset of $\mathcal{A}_{s,c}$ contained
in the union $\mathcal{U}_{min}$ or $\mathcal{U}_{\neg min}$).

####  2.1.3. Ergodicity and reducibility of extensions

Let $\Phi^{f}_{\mathbb{R}}:=(\Phi^{f}_{t})_{t\in\mathbb{R}}$ on
$M\times{\mathbb{R}}$ denotes the extension of an ergodic flow
$\psi_{\mathbb{R}}$ on $M$ by $f:M\to{\mathbb{R}}$ given by the formula (1.3).
Recall that, if $\psi_{\mathbb{R}}$ preserves a measure $\mu$,
$\Phi^{f}_{\mathbb{R}}$ preserves the (infinite) measure $\mu\times Leb$. The
flow $\Phi^{f}_{\mathbb{R}}$ is _recurrent_ if $\mu\times Leb$-almost every
point is recurrent. A result by Atkinson [3] (which holds for $1$-dimensional
extensions of ergodic flows) shows that $\Phi^{f}_{\mathbb{R}}$ is recurrent
if and only if $\int_{M}f\mathrm{d}\mu=0$.

We recall that $\Phi^{f}_{{\mathbb{R}}}$ is _ergodic_ with respect to the
(infinite) measure $\mu\times Leb$ if for any measurable set $A$ which is
invariant, i.e. such that $\mu\times Leb(A)=\mu\times Leb(\Phi^{f}_{t}A)$ for
all $t\in\mathbb{R}$, either $\mu\times Leb(A)=0$ or $\mu\times Leb(A^{c})=0$,
where $A^{c}$ denotes the complement.

Remark that if $f=0$, the phase space $M\times{\mathbb{R}}$ for the
corresponding trivial extension given by $\Phi^{f}_{t}(x,y)=(\psi_{t}(x),y)$
is foliated in invariant sets of the form $M\times\\{y\\}$,
$y\in{\mathbb{R}}$. In this sense, the dynamics is reduced to the dynamics of
the surface flow $\psi_{\mathbb{R}}$. We say that $\Phi^{f}_{\mathbb{R}}$ is
(topologically) reducible if it is isomorphic to $\Phi^{0}_{\mathbb{R}}$ and
the isomorphism $G:M\times{\mathbb{R}}\to M\times{\mathbb{R}}$ is of the form
$G(x,y)=(x,y+g(x))$, where $g:M\to{\mathbb{R}}$ is continuous. So the
reducibility of $\Phi^{f}_{\mathbb{R}}$ is equivalent to asking that

$\int_{0}^{t}f(\psi_{s}x)\,ds=g(x)-g(\psi_{t}x)$

for every regular point $x\in M$ and any $t\in{\mathbb{R}}$. In this case, the
phase space is again foliated into invariant sets for $\Phi^{f}_{\mathbb{R}}$
of the form $\\{(x,y+g(x)),\ x\in M\\}$, $\ y\in{\mathbb{R}}$. On each leaf
the action of $\Phi^{f}_{{\mathbb{R}}}$ is conjugated to $\psi_{{\mathbb{R}}}$
on $M$.

####  2.1.4. Typical chaotic properties of locally Hamiltonian flows

Let us briefly summarize the key chaotic properties of locally Hamiltonian
flows and some of the recent works on this topic. We already recalled in the
introduction, in view of the relation between locally Hamiltonian flows and
translation flows (see also Remark 2.3), the seminal works by Keane [36] and
Masur [49] and Veech [64] show that a full measure set of locally Hamiltonian
flows in $\mathcal{U}_{min}$ are minimal and ergodic and that almost every
flow in $\mathcal{U}_{\neg min}$, the restriction to each minimal component is
ergodic (and in both cases the underlying foliation in uniquely ergodic).

Mixing depends crucially on the type of singularities of the flow. For a (non-
generic) locally Hamiltonian flow with at least one _degenerate saddle_ (see
e.g. Figure 2(c)), mixing was proved in the $1970s$ (by Kochergin in [39]).
When $\eta\in\mathcal{A}$ and all saddles are _simple_ , one has the following
dichotomy: in $\mathscr{U}_{min}$, the typical locally Hamiltonian flow is
_weakly mixing, but it is _not_ mixing_ in view of work [62, 63] by the second
author (see also [37, 38] and [58] for previous special cases of this result).
There exist nevertheless exceptional mixing flows, see the work by [8], which
produces sporadic examples in $g=5$. If $\eta\in\mathcal{U}_{\neg min}$, the
restriction of the typical locally Hamiltonian flow $\psi_{\mathbb{R}}$ on
each of its minimal components is mixing (as proved by Ravotti [56] extending
previous work by the second author [61]). Ravotti also shows in [56]
_subpolynomial_ bounds for the speed of mixing.

Further recent work (see [30]) also shows that locally Hamiltonian flows in
$\mathcal{U}_{\neg min}$ display a _quantitative shearing_ property inspired
by the _Ratner property_ which plays a crucial role in the theory of unipotent
flows (or more precisely a variation introduced in [13] to deal with the
presence of singularities). From this property, one can deduce that the
restriction of a typical locally Hamiltonian flow $\psi_{\mathbb{R}}$ in
$\mathscr{U}_{\neg min}$ on its minimal components is not only mixing, but
_mixing of all orders_ , see [30]. Arnold flows in genus one were also
recently shown (by A. Kanigowski and M. Lemańczyk and the second author, see
[31]) to typically have _disjointess 161616The notion of _disjointness_ in
ergodic theory was introduced in the $1970s$ by H. Furstenberg, see in
particular [26]. of rescalings_, a property which in particular implies Sarnak
Möbius orthogonality conjecture [57] to hold (see [31] for details and [15]
for a nice survey on the conjecture and progress toward it).

The spectral theory of locally Hamiltonian flows is still largely not
understood. Examples171717These examples are known as _Blokhin examples_ and
are essentially built glueing genus one flows. This allows to study them using
(special flows over) rotations. On the other hand, they are highly non
typical. of locally Hamiltonian flows on surfaces of any genus $\geq 1$ with
singular continuous spectrum were build by M. Lemańczyk and the first author
(see [19, Theorem 1]). For some flows in genus one with a _degenerate_
singularity (sometimes known as _Kochergin flows_), Forni, Fayad and
Kanigowski could recently, prove in [12] that the spectrum is _countably
Lebesgue_. The first typical spectral result for surfaces of higher genus,
namely $g\geq 2$ was recently proved by Chaika, Kanigowski and the authors,
who showed in [9] that a typical locally Hamiltonian flow on a _genus two_
surface with two isomorphic simple saddles has _purely singular_ spectrum.

###  2.2. IETs, special flows and extension.

Let us now introduce the notation that we will use for interval exchange
transformations (§ 2.2.1) and recall the definition of two basic
constructions, special flows (§ 2.2.2) and extensions of IETs (§ 2.2.3).

####  2.2.1. Interval exchange transformations.

Let $\mathcal{A}$ be a $d$-element alphabet and let $\pi=(\pi_{0},\pi_{1})$ be
a pair of bijections $\pi_{\varepsilon}:\mathcal{A}\to\\{1,\ldots,d\\}$ for
$\varepsilon=0,1$. We adopt the notation from [66]. Denote by
$\mathcal{S}^{0}_{\mathcal{A}}$ the subset of irreducible pairs, i.e. such
that $\pi_{1}\circ\pi_{0}^{-1}\\{1,\ldots,k\\}\neq\\{1,\ldots,k\\}$ for $1\leq
k<d$.

For any
$\lambda=(\lambda_{\alpha})_{\alpha\in\mathcal{A}}\in{\mathbb{R}}_{>0}^{\mathcal{A}}$
let

$|\lambda|=\sum_{\alpha\in\mathcal{A}}\lambda_{\alpha},\quad
I=\left[0,|\lambda|\right)$

and define

$I_{\alpha}=[l_{\alpha},r_{\alpha}),\text{ where
}l_{\alpha}=\sum_{\pi_{0}(\beta)<\pi_{0}(\alpha)}\lambda_{\beta},\;\;\;r_{\alpha}=\sum_{\pi_{0}(\beta)\leq\pi_{0}(\alpha)}\lambda_{\beta}.$

Then $|I_{\alpha}|=\lambda_{\alpha}$. Denote by $\Omega_{\pi}$ the matrix
$[\Omega_{\alpha\,\beta}]_{\alpha,\beta\in\mathcal{A}}$ given by

$\Omega_{\alpha\,\beta}=\left\\{\begin{array}[]{cl}+1&\text{ if
}\pi_{1}(\alpha)>\pi_{1}(\beta)\text{ and }\pi_{0}(\alpha)<\pi_{0}(\beta),\\\
-1&\text{ if }\pi_{1}(\alpha)<\pi_{1}(\beta)\text{ and
}\pi_{0}(\alpha)>\pi_{0}(\beta),\\\ 0&\text{ in all other
cases.}\end{array}\right.$

Given
$(\pi,\lambda)\in\mathcal{S}^{0}_{\mathcal{A}}\times{\mathbb{R}}_{>0}^{\mathcal{A}}$
let $T_{(\pi,\lambda)}:[0,|\lambda|)\rightarrow[0,|\lambda|)$ stand for the
interval exchange transformation (IET) on $d$ intervals $I_{\alpha}$,
$\alpha\in\mathcal{A}$, which are rearranged according to the permutation
$\pi^{-1}_{1}\circ\pi_{0}$, i.e. $T_{(\pi,\lambda)}x=x+w_{\alpha}$ for $x\in
I_{\alpha}$, where $w=\Omega_{\pi}\lambda$.

Keane condition. Let $End(T)$ stand for the set of end points of the intervals
$I_{\alpha}:\alpha\in\mathcal{A}$. A pair ${(\pi,\lambda)}$ satisfies the
Keane condition if $T_{(\pi,\lambda)}^{m}l_{\alpha}\neq l_{\beta}$ for all
$m\geq 1$ and for all $\alpha,\beta\in\mathcal{A}$ with $\pi_{0}(\beta)\neq
1$. Keane [36] showed that an IET with an irreducible permutation that satisfy
the Keane condition is _minimal_.

We record here two remarks that will be useful later.

######  Remark 2.1.

Note that for every $\alpha\in\mathcal{A}$ with $\pi_{0}(\alpha)\neq 1$ there
exists $\beta\in\mathcal{A}$ such that $\pi_{0}(\beta)\neq d$ and
$l_{\alpha}=r_{\beta}$. It follows that

$\\{l_{\alpha}:\alpha\in\mathcal{A},\;\pi_{0}(\alpha)\neq
1\\}=\\{r_{\alpha}:\alpha\in\mathcal{A},\;\pi_{0}(\alpha)\neq d\\}.$

######  Remark 2.2.

Denote by $\widehat{T}_{(\pi,\lambda)}:(0,|I|]\to(0,|I|]$ the exchange of the
intervals $\widehat{I}_{\alpha}:=(l_{\alpha},r_{\alpha}]$,
$\alpha\in\mathcal{A}$, i.e. $T_{(\pi,\lambda)}x=x+w_{\alpha}$ for
$x\in(l_{\alpha},r_{\alpha}]$. Note that for every $\alpha\in\mathcal{A}$ with
$\pi_{1}(\alpha)\neq 1$ there exists $\beta\in\mathcal{A}$ such that
$\pi_{1}(\beta)\neq d$ and
$T_{(\pi,\lambda)}l_{\alpha}=\widehat{T}_{(\pi,\lambda)}r_{\beta}$.

####  2.2.2. Special flow definition

Let $T:I\to I$ be an (ergodic) IET and let
$r:I\to{\mathbb{R}}_{>0}\cup\\{+\infty\\}$ be an integrable function such that
$\underline{r}=\inf_{x\in I}r(x)>0$. The _special flow_ over $T$ under the
_roof function_ $r$ is the flow
$T^{r}_{\mathbb{R}}:=(T^{r}_{t})_{t\in{\mathbb{R}}}$ acting on

$I^{r}:=\\{(x,s)\in I\times{\mathbb{R}}:0\leq s<r(x)\\},$

so that $T^{r}_{t}(x,s)=(x,s+t-r^{(n)}(x))$, where $r^{(n)}(x)$ denote the
Birkhoff sums cocycle181818Here $r^{(n)}(x)$ denotes the additive cocycle
defined by $r^{(n)}(x):=\sum_{0\leq k<n}r(T^{k}x)$ if $n\geq 0$ and
$r^{(n)}(x):=-\sum_{n\leq k<0}r(T^{k}(x))$ if $n<0$. associated to $r$ and $n$
is the unique integer number with $r^{(n)}(x)\leq s+t<r^{(n+1)}(x)$. It
describes the motion of a point in $(x,s)\in I^{r}\subset I\times{\mathbb{R}}$
along vertical trajectories, modulo the identification of each point
$(x,r(x))$, $x\in I$, with the point $(Tx,0)$.

####  2.2.3. Skew product extensions

Given an IET $T:I\to I$ and a function $\varphi:I\to{\mathbb{R}}$ the
_extension_ of $T$ by $\varphi$ is the skew-product map
$T_{\varphi}:I\times{\mathbb{R}}\to I\times{\mathbb{R}}$ defined as in (1.4)
by $T_{\varphi}(x,y)=(T(x),y+\varphi(x))$. Notice that, for $n\geq 0$, the
iterates of $T_{\varphi}$ have the form

$T^{n}_{\varphi}(x,y)=(T^{n}(x),y+\varphi^{(n)}(x)),\qquad\text{where}\quad\varphi^{(n)}(x):=\sum_{k=0}^{n-1}\varphi(T^{k}(x)).$

Remark that the Birkhoff sums $\varphi^{(n)}(\cdot)$ are a (additive)
_cocycle_ over $T$ in view of the _cocycle relation_
$\varphi^{(m+n)}(x)=\varphi^{(m)}(T^{n}x)+\varphi^{(n)}(x)$.

###  2.3. Reduction to special flows and skew-product presentations

We recall two classical results that show that locally Hamiltonian flows and
their extensions can be reduced respectively to the study of special flows and
skew-product extensions over IETs, with roof functions or, respectively,
cocycles, with logarithmic singularities.

####  2.3.1. Logarithmic singularities

We say that a function (or cocycle) $\varphi:I\to{\mathbb{R}}$ for an IET
$T_{(\pi,\lambda)}$ has _logarithmic singularities_ if there exist constants
$C_{\alpha}^{+},C_{\alpha}^{-}\in\mathbb{R}$, $\alpha\in\mathcal{A}$, and a
function $g_{\varphi}$ absolutely continuous on the interior of each interval
$I_{\alpha}$, $\alpha\in\mathcal{A}$ (i.e. with the notation that we will
introduce later, a function
$g_{\varphi}\in\operatorname{AC}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$)
such that

(2.1) $\displaystyle\varphi(x)=$
$\displaystyle-\sum_{\alpha\in\mathcal{A}}C^{+}_{\alpha}\log\big{(}|I|\\{(x-l_{\alpha})/|I|\\}\big{)}-\sum_{\alpha\in\mathcal{A}}C^{-}_{\alpha}\log\big{(}|I|\\{(r_{\alpha}-x)/|I|\\}\big{)}+g_{\varphi}(x).$

We refer to Figure 4 for some examples. We say that the logarithmic
singularities are _of geometric type_ if at least one among
$C_{\pi_{0}^{-1}(d)}^{-}$ and $C_{\pi_{1}^{-1}(d)}^{-}$ is zero and at least
one among $C_{\pi_{0}^{-1}(1)}^{+}$ or $C_{\pi_{1}^{-1}(1)}^{+}$ is zero (as
shown in the examples in Figure 4). We denote by
$\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the space of
functions with logarithmic singularities of geometric type. We define also the
subspace $\operatorname{LSG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
$\subset\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ of
functions satisfying the symmetry condition

(2.2)
$\sum_{\alpha\in\mathcal{A}}C^{-}_{\alpha}-\sum_{\alpha\in\mathcal{A}}C^{+}_{\alpha}=0.$

(a) Roof function
$r\in\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$

(b) Cocycle
$\varphi\in\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$

Figure 4. Examples of functions with geometric logarithmic singularities in
$\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$.

####  2.3.2. Special flow representations of locally Hamiltonian flows

It is well known that locally Hamiltonian flows can be represented as special
flows as follows (see for example [63, 56, 10, 20]). Consider either a minimal
locally Hamiltonian flow $\psi_{\mathbb{R}}$ on $M$ or the restriction of a
locally Hamiltonian flow on $M$ to a minimal component $M^{\prime}\subset M$.
Let $\eta$ be the associated closed $1$-form and assume that
$\eta\in\mathcal{A}$, i.e. $\eta$ is Morse. Then $\psi_{\mathbb{R}}$ can be
shown to be (measure theoretically) _isomorphic_ to a _special flow_
$T^{r}:I^{r}\to I^{r}$ over an interval exchange transformation $T:I\to I$ of
$d\geq 1$ intervals and under a roof
$r\in\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$. The number
of exchanged intervals is $d=2g+s-1$ in the case when $\psi_{\mathbb{R}}$ is
minimal and $s$ is the number of simple saddles, or, for a minimal component
$M^{\prime}$, $d=2g^{\prime}+s^{\prime}-1$, where $g^{\prime}$ is the genus of
$M^{\prime}$ and $s^{\prime}$ is the number of saddles in the closure of
$M^{\prime}$. Furthermore, if $\eta\in\mathcal{U}_{min}$, the logarithmic
singularities are _symmetric_ , i.e.
$\varphi\in\operatorname{LSG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ (while
they are _asymmetric_ for special flows representations of minimal components
of typical $\eta\in\mathcal{U}_{\neg min}$).

######  Remark 2.3.

We recall for contrast that also _translation flows_ can be seen as special
flows over an interval exchange map, but under a roof function $r$ which is
_piecewise-constant_ (and constant on each continuity interval of the IET).
One can therefore see from these special representations that minimal
(components of) locally Hamiltonian flows are time-changes of translation
flows via a _singular_ reparametrization.

####  2.3.3. Reduction to skew products.

The study of (ergodic properties of) extensions can be reduced to the study of
skew-products over IETs as follows.

######  Proposition 2.4 (Reduction of ergodicity of extensions to skew
products).

Consider a Morse closed one-form $\eta\in\mathcal{A}$ on $M$ and let
$\psi_{\mathbb{R}}$ on $M$ be the associated locally Hamiltonian flow.
Consider its minimal component $M^{\prime}\subset M$. For every
$C^{2+\epsilon}$-map $f:M^{\prime}\to{\mathbb{R}}$ ($\epsilon>0$), the
extension $\Phi^{f}_{\mathbb{R}}$ of $\psi_{\mathbb{R}}$ on $M^{\prime}$ has a
Poincaré map which, in suitable coordinates, is given by a skew-product of the
form

(2.3) $(x,y)\mapsto T_{\varphi_{f}}(x,y):=(Tx,y+\varphi_{f}(x)),\qquad(x,y)\in
I\times{\mathbb{R}}.$

where $T=T_{(\pi,\lambda)}$ with $\pi$ irreducible and the cocycle
$\varphi_{f}:I\to{\mathbb{R}}$ has logarithmic singularities, i.e
$\varphi_{f}\in\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
where $(I_{\alpha})_{\alpha\in\mathcal{A}}$ are intervals exchanged by $T$.

Moreover, the extension $\Phi^{f}_{\mathbb{R}}$ on
$M^{\prime}\times{\mathbb{R}}$ is ergodic with respect to $\mu\times Leb$ if
and only if $T_{\varphi_{f}}:I\times{\mathbb{R}}\to I\times{\mathbb{R}}$ is
ergodic with respect to the (restriction of) the $2$-dimensional Lebesgue
measure on $I\times{\mathbb{R}}$.

We give here only a brief sketch of the proof, referring to the proof in [20]
for details.

###### Proof.

Fix a segment $\gamma\subset M^{\prime}\subset M$ transverse to the flow
$\psi_{\mathbb{R}}$, containing no fixed points and whose endpoints lie on
outgoing separatrices of saddles. It is well known (see for example [68,
Section 4.4]) that one can choose a parametrization $t\in I\to\gamma(t)$ of
$\gamma$ by the unit interval $I=[0,1)$ so that the Poincaré first return map
$T:I\to I$ of the flow $\psi_{\mathbb{R}}$ to $\gamma$ is an IET, which is
minimal by assumption. It follows that $\pi$ is irreducible.

Denote by $r:I\to{\mathbb{R}}_{>0}$ the first return time map for the flow
$(\psi_{t})_{t\in{\mathbb{R}}}$ on $M^{\prime}$. Then the isomorphism between
the restriction of $\psi_{\mathbb{R}}$ to $M^{\prime}$ and a special flow
$T^{r}$ on $I^{r}$ is given by

$I^{r}\ni(x,r)\mapsto\psi_{r}(x)\in M^{\prime}.$

As recalled in the previous § 2.3.2,
$r\in{\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ and
moreover, if $\psi_{\mathbb{R}}\in\mathcal{U}_{min}$, i.e. $M^{\prime}=M$,
then $r\in{\operatorname{LSG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$, see
e.g. [56].

Consider now the extension $\Phi^{f}_{\mathbb{R}}$ of $\psi_{\mathbb{R}}$ on
$M^{\prime}$ given by a bounded function $f:M^{\prime}\to{\mathbb{R}}$. The
Poincaré map of $\Phi^{f}_{\mathbb{R}}$ on $M^{\prime}\times{\mathbb{R}}$ to
the section $\gamma\times{\mathbb{R}}$ in the parametrization by
$I\times{\mathbb{R}}$ is by construction an extension of the Poincaré map $T$
of $\psi_{\mathbb{R}}$ to $I$, with return time function $r(x,y)=r(x)$ (i.e.
the return time only depends on the return to $I$ in the first coordinate, by
definition of the section which has full fiber). Moreover, if we consider the
cocycle

(2.4) $\varphi_{f}(x):=\int_{0}^{r(x)}f(\psi_{t}(x))\,dt$

(which gives the value of the ergodic integrals of $f$ along the trajectory
from $x$ until the first return time to the section), one can then see that
the first return Poincaré map of the extension $\Phi^{f}_{\mathbb{R}}$ has the
form (2.3). If $f$ is a $C^{2+\epsilon}$-map, from the explicit expression
(2.4) and the properties of $r$, one can then show that also
$\varphi_{f}\in\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
(see [20] for details) and
$\varphi_{f}\in{\operatorname{LSG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
if $\eta\in\mathcal{U}_{min}$.

The final statement is simply a consequence that ergodicity of a minimal flow
is equivalent to ergodicity of its Poincaré map with respect to the induced
measure, together with the remark that, under the isomorphism described above,
the measure induced on the section $\gamma\times{\mathbb{R}}$ by the invariant
measure $\mu\times Leb$ is mapped to the Lebesgue measure on
$I\times{\mathbb{R}}$. ∎

The following result shows that not only ergodicity, but also reducibility of
the extension $\Phi^{f}_{\mathbb{R}}$ can be reduced to a property of the skew
product $T_{\varphi_{f}}$ given by Proposition 2.4.

######  Proposition 2.5 (Reduction of reducibility to skew products, [20]).

For every minimal locally Hamiltonian flow $\psi_{\mathbb{R}}$ on $M$ with
non-degenerate saddles and any $f\in C^{2+\epsilon}(M)$ vanishing on
$\mathrm{Fix}(\psi_{\mathbb{R}})$, the associated flow $\Phi^{f}_{\mathbb{R}}$
is reducible if and only if the cocycle $\varphi_{f}:I\to{\mathbb{R}}$ is a
coboundary with a bounded transfer map having at least one continuity point,
i.e. there exists a bounded $g:I\to{\mathbb{R}}$ such that
$\varphi_{f}=g-g\circ T$ and $g$ has at least one continuity point.

The statement of the Proposition is proved in the proof191919Note that the
statement of Lemma 6.3 in [20] claims incorrectly that reducibility requires
the existence of transfer function continous at _every_ point, while a the
_existence_ of a point of continuity is sufficient. Nevertheless, the _proof_
of Lemma 6.3 in [20] is correct and gives a proof of the statement of
Proposition 2.5 here above. of Lemma 6.3 in [20].

##  3\. Rauzy-Veech induction and Diophantine-type conditions

In this section we define the Diophantine-type condition on IETs which we will
use to prove our main results on deviations of ergodic averages and ergodicity
of extensions. The condition is described in terms of Rauzy-Veech induction,
an algorithm introduced by Rauzy and Veech in [55, 64] which is now a well
established tool to study IETs as well to impose Diophantine conditions on
them (see e.g. [4, 5, 6, 44, 46, 63, 61, 70] and many more). We first recall
some basic background material concerning Rauzy-Veech induction in § 3.1. The
condition, that we call _Uniform Diophantine Condition_ , or for short
LABEL:UDC, is defined in § 3.2 (see Definition 3 in § 3.2.2). In § 3.2.3 we
also prove that this condition is satisfied by a full measure set of IETs (see
Theorem 3.8).

###  3.1. Rauzy-Veech induction

We recall here some basic definitions and notation related to Rauzy-Veech
induction that will be used throughout the paper, including how it acts on
Rokhlin towers (§ 3.1.4) and on Birkhoff sums (§ 3.1.5), as well as the
definition of natural extension (§ 3.1.6). We recall also Oseledets theorem (§
3.1.7).

####  3.1.1. Elementary step of RV induction

Let $T=T_{(\pi,\lambda)}$,
$(\pi,\lambda)\in\mathcal{S}^{0}_{\mathcal{A}}\times{\mathbb{R}}_{>0}^{\mathcal{A}}$
be an IET satisfying Keane’s condition. Then
$\lambda_{\pi_{0}^{-1}(d)}\neq\lambda_{\pi_{1}^{-1}(d)}$. Let

$\tilde{I}=\left[0,\max\left({l}_{\pi_{0}^{-1}(d)},{l}_{\pi_{1}^{-1}(d)}\right)\right)$

and denote by $\mathcal{R}(T)=\tilde{T}:\tilde{I}\to\tilde{I}$ the first
return map of $T$ to the interval $\tilde{I}$. Set

$\varepsilon(\pi,\lambda)=\left\\{\begin{array}[]{ccl}0&\text{ if
}&\lambda_{\pi_{0}^{-1}(d)}>\lambda_{\pi_{1}^{-1}(d)},\\\ 1&\text{ if
}&\lambda_{\pi_{0}^{-1}(d)}<\lambda_{\pi_{1}^{-1}(d)}.\end{array}\right.$

Let us consider a pair
$\tilde{\pi}=(\tilde{\pi}_{0},\tilde{\pi}_{1})\in\mathcal{S}^{0}_{\mathcal{A}}$,
where

$\displaystyle\tilde{\pi}_{\varepsilon}(\alpha)$ $\displaystyle=$
$\displaystyle\pi_{\varepsilon}(\alpha)\text{ for all
}\alpha\in\mathcal{A}\text{ and }$
$\displaystyle\tilde{\pi}_{1-\varepsilon}(\alpha)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cll}\pi_{1-\varepsilon}(\alpha)&\text{
if
}&\pi_{1-\varepsilon}(\alpha)\leq\pi_{1-\varepsilon}\circ\pi^{-1}_{\varepsilon}(d),\\\
\pi_{1-\varepsilon}(\alpha)+1&\text{ if
}&\pi_{1-\varepsilon}\circ\pi^{-1}_{\varepsilon}(d)<\pi_{1-\varepsilon}(\alpha)<d,\\\
\pi_{1-\varepsilon}\pi^{-1}_{\varepsilon}(d)+1&\text{ if
}&\pi_{1-\varepsilon}(\alpha)=d.\end{array}\right.$

As it was shown by Rauzy in [55], $\tilde{T}$ is also an IET on $d$-intervals

(3.2) $\tilde{T}=T_{(\tilde{\pi},\tilde{\lambda})}\text{ with
}\tilde{\lambda}=A^{-1}(\pi,\lambda)\lambda,$

where

$A(T)=A(\pi,\lambda)=I+E_{\pi_{\varepsilon}^{-1}(d)\,\pi_{1-\varepsilon}^{-1}(d)}\in\operatorname{SL}({\mathbb{Z}}^{\mathcal{A}}).$

Moreover,

(3.3) $A^{t}(\pi,\lambda)\Omega_{\pi}A(\pi,\lambda)=\Omega_{\tilde{\pi}}.$

It follows that $\ker\Omega_{{\pi}}=A(\pi,\lambda)\ker\Omega_{\tilde{\pi}}$.
Thus taking
$H(\pi)=\Omega_{{\pi}}({\mathbb{R}}^{\mathcal{A}})=\ker\Omega_{{\pi}}^{\perp}$
we get

(3.4) $H(\tilde{\pi})=A^{t}(\pi,\lambda)H(\pi).$

Moreover, $\dim H(\pi)=2g$ and $\dim\ker\Omega_{{\pi}}=\kappa(\pi)-1$, where
$\kappa(\pi)$ is the number of singularities and $g$ is the genus of the
translation surfaces associated to $\pi$.

####  3.1.2. Renormalized induction

Let $\mathcal{G}\subset\mathcal{S}^{0}_{\mathcal{A}}$ be any Rauzy class, i.e.
a minimal subset of $\mathcal{S}^{0}_{\mathcal{A}}$ for which
$\mathcal{G}\times{\mathbb{R}}_{>0}^{\mathcal{A}}$ is $\mathcal{R}$-invariant.
Let

$\Delta^{\mathcal{A}}:=\\{\lambda\in{\mathbb{R}}_{>0}^{\mathcal{A}}:|\lambda|=1\\}.$

Then we can define the normalized Rauzy-Veech renormalization

$\widetilde{\mathcal{R}}:\mathcal{G}\times\Delta^{\mathcal{A}}\to\mathcal{G}\times\Delta^{\mathcal{A}},\quad\widetilde{\mathcal{R}}(\pi,\lambda)=(\tilde{\pi},\tilde{\lambda}/|\tilde{\lambda}|).$

Veech in [64] proved the existence of an $\widetilde{\mathcal{R}}$-invariant
ergodic measure $\mu_{\mathcal{G}}$ ($\widetilde{\mathcal{R}}$ is recurrent
with respect to $\mu_{\mathcal{G}}$) which is equivalent to the product of the
counting measure on $\mathcal{G}$ and the Lebesgue measure on
$\Delta^{\mathcal{A}}$.

For every $T$ satisfying the Keane condition, the IET $\tilde{T}$ fulfills the
Keane condition as well. Therefore we can iterate the renormalization
procedure and generate a sequence of IETs $(\mathcal{R}^{n}(T))_{n\geq 0}$.
For every $n\geq 1$ let

$A^{(n)}(T)=A(T)\cdot A(\mathcal{R}(T))\cdot\ldots\cdot
A(\mathcal{R}^{n-1}(T)).$

In what follows, the norm of a vector is defined as the sum of the absolute
value of coefficients and for any matrix
$B=[B_{\alpha\beta}]_{\alpha,\beta\in\mathcal{A}}$ we set
$\|B\|=\max_{\alpha\in\mathcal{A}}\sum_{\beta\in\mathcal{A}}|B_{\alpha\beta}|$.

####  3.1.3. Accelerations.

Let $T:I\to I$ be an arbitrary IET satisfying Keane’s condition. Let
$(n_{k})_{k\geq 0}$ be an increasing sequence of integer numbers with
$n_{0}=0$, called an _accelerating sequence_. For every $k\geq 0$ let
$T^{(k)}:=\mathcal{R}^{n_{k}}(T):I^{(k)}\to I^{(k)}$. Denote by
$(\pi^{(k)},\lambda^{(k)})$ the pair defining $T^{(k)}$ and by
$\lambda^{(k)}=(\lambda^{(k)}_{\alpha})_{\alpha\in\mathcal{A}}=(|I^{(k)}_{\alpha}|)_{\alpha\in\mathcal{A}}$
the vector which determines $T^{(k)}$.

In view of (3.2), letting
$Z(k+1):=A^{(n_{k+1}-n_{k})}(\mathcal{R}^{n_{k}}(T))^{t}$ for $k\geq 0$ we
have

$\lambda^{(k)}=Z(k+1)^{t}\lambda^{(k+1)}\text{ for all }k\geq 0.$

We use the notation from [44], but adopt the convention later introduced in
[48]. For each $0\leq k<l$ let

$Q(k,l)=Z(l)\cdot Z(l-1)\cdot\ldots\cdot Z(k+2)\cdot
Z(k+1)=A^{(n_{l}-n_{k})}(\mathcal{R}^{n_{k}}(T))^{t}.$

Then $Q(k,l)\in SL_{\mathcal{A}}({\mathbb{Z}})$ and

$\lambda^{(k)}=Q(k,l)^{t}\lambda^{(l)}.$

It follows that

(3.5) $|I^{(k)}|\leq|I^{(l)}|\|Q(k,l)\|.$

We will write $Q(k)$ for $Q(0,k)$.

We say that $Z(k)$, $k\in{\mathbb{N}}$ (resp. $Q(k,l)$) are the _matrices_
(resp. the _product matrices_) of the acceleration of $A$ along the
(accelerating) sequence $(n_{k})_{k\in{\mathbb{N}}}$

####  3.1.4. Rokhlin towers

By definition, $T^{(l)}:I^{(l)}\to I^{(l)}$ is the first return map of
$T^{(k)}:I^{(k)}\to I^{(k)}$ to the interval $I^{(l)}\subset I^{(k)}$.
Moreover, $Q_{\alpha\beta}(k,l)$ is the time spent by any point of
$I^{(l)}_{\alpha}$ in $I^{(k)}_{\beta}$ until it returns to $I^{(l)}$. It
follows that

$Q_{\alpha}(k,l)=\sum_{\beta\in\mathcal{A}}Q_{\alpha\beta}(k,l)$

is the first return time of points of $I^{(l)}_{\alpha}$ to $I^{(l)}$.

The map $T^{(k)}:I^{(k)}\to I^{(k)}$ can be then represented as a _Rokhlin
skyscraper_ as follows. For every $\alpha\in\mathcal{A}$, we say that the set

$\\{(T^{(k)})^{i}(I^{(l)}_{\alpha}),\qquad 0\leq i<Q_{\alpha}(k,l)\\}$

is called a _Rokhlin tower_. Notice that the $Q_{\alpha}(k,l)$ sets part of it
are _disjoint_ intervals called _floors_ of the tower and that, for $0\leq
i<Q_{\alpha}(k,l)$, $T^{(k)}$ acts on the $i^{th}$ floor
$(T^{(k)})^{i}(I^{(l)}_{\alpha})$ mapping it to the $(i+1)^{th}$ one. The
union of all Rokhlin towers over $\alpha\in\mathcal{A}$ gives $I^{(k)}$.

####  3.1.5. Special Birkhoff sums.

We deal with the _special Birkhoff sums_ operators $S(k,l):L^{1}(I^{(k)})\to
L^{1}(I^{(l)})$ for $0\leq k<l$ defined by

$S(k,l)f(x)=\sum_{0\leq j<Q_{\alpha}(k,l)}f((T^{(k)})^{j}x)\quad\text{ if
}\quad x\in I^{(l)}_{\alpha}.$

Let $T=T^{(0)}$ be an IET satisfying Keane’s condition. For every $k\geq 0$
let $\Gamma^{(k)}\subset L^{1}(I^{(k)})$ be the subspace of functions on
$I^{(k)}$ which are constant on each $I^{(k)}_{\alpha}$,
$\alpha\in\mathcal{A}$. Then for $0\leq k<l$ we have
$S(k,l)\Gamma^{(k)}=\Gamma^{(l)}$. Let us identify every function
$\sum_{\alpha\in\mathcal{A}}h_{\alpha}\chi_{I^{(k)}_{\alpha}}\in\Gamma^{(k)}$
with the vector
$h=(h_{\alpha})_{\alpha\in\mathcal{A}}\in{\mathbb{R}}^{\mathcal{A}}$. Clearly
$\Gamma^{(k)}$ is isomorphic to $\mathbb{R}^{\mathcal{A}}$. Under the
identification, the operator $S(k,l)$ is the linear automorphism of
${\mathbb{R}}^{\mathcal{A}}$ whose matrix in the canonical basis is $Q(k,l)$.
In view of (3.4) for $0\leq k<l$ we have

$Q(k,l)H(\pi^{(k)})=H(\pi^{(l)}).$

For every $k\geq 0$ let

$\displaystyle\Gamma^{(k)}_{s}:=\\{h\in\Gamma^{(k)}:\exists_{\sigma>0}\exists_{C>0}\forall_{l>k}\,\|Q(k,l)h\|\leq
C\|Q(k,l)\|^{-\sigma}\\}.$

The space $\Gamma^{(k)}_{s}$ is a subspace of $H(\pi^{(k)})$ and for every
$l>k$ we have

$Q(k,l)\Gamma^{(k)}_{s}=\Gamma^{(l)}_{s}.$

Therefore, the restriction operator and the quotient operators of $Q(k,l)$

$\displaystyle Q_{s}(k,l):\Gamma^{(k)}_{s}\to\Gamma^{(l)}_{s},\quad
Q_{\flat}(k,l):\Gamma^{(k)}/\Gamma^{(k)}_{s}\to\Gamma^{(l)}/\Gamma^{(l)}_{s},\quad
Q_{\sharp}(k,l):H(\pi^{(k)})/\Gamma^{(k)}_{s}\to
H(\pi^{(l)})/\Gamma^{(l)}_{s}$

are well defined and are invertible. Arguments presented in Section 3.2 in
[48] shows that if $\dim\Gamma^{(0)}_{s}=g$ then

(3.6) $\|Q_{\sharp}(k,l)^{-1}\|=\|Q_{s}(k,l)\|.$

####  3.1.6. The natural extension

Rauzy-Veech induction is not intertible, but it can be extended to an
invertible induction on the space of _zippered rectangles_ (as described in
the seminar paper by Veech [64]). We recall briefly the construction. We refer
the reader who needs more background to the lecture notes by Yoccoz [68] or
Viana [66].

For every $\pi\in S^{\mathcal{A}}_{0}$ let

$\Theta_{\pi}:=\Big{\\{}\tau\in{\mathbb{R}}^{\mathcal{A}}:\sum_{\pi_{0}(\alpha)\leq
k}\tau_{\alpha}>0,\;\sum_{\pi_{1}(\alpha)\leq k}\tau_{\alpha}<0\text{ for
}1\leq k<d\Big{\\}}.$

For every $\tau\in\Theta_{\pi}$ let
$h=h(\tau)=\Omega\tau\in{\mathbb{R}}^{\mathcal{A}}_{>0}$. For every Rauzy
class $\mathcal{G}\subset S^{\mathcal{A}}_{0}$ let

(3.7)
$X(\mathcal{G})=\bigcup_{\pi\in\mathcal{G}}\\{(\pi,\lambda,\tau)\in\\{\pi\\}\times\Delta^{\mathcal{A}}\times\Theta_{\pi}:\langle\lambda,\Omega_{\pi}\tau\rangle=1\\}.$

For every $(\pi,\lambda,\tau)\in X(\mathcal{G})$ denote by
$M(\pi,\lambda,\tau)$ the translation surface arising in the zippered
rectangles process. Then $M(\pi,\lambda,\tau)$ is zippered from the rectangles
$I_{\alpha}\times[0,h_{\alpha}]$, $\alpha\in\mathcal{A}$ such that the points
$\sum_{\pi_{0}(\alpha)\leq k}(\lambda_{\alpha}+i\tau_{\alpha})$, $0\leq k\leq
d$ are its singular points. Moreover, the IET $T$ is the first return map to
$I\subset M(\pi,\lambda,\tau)$ for the vertical flow on $M(\pi,\lambda,\tau)$.

The map $\widehat{\mathcal{R}}:X(\mathcal{G})\to X(\mathcal{G})$ given by

$\widehat{\mathcal{R}}(\pi,\lambda,\tau)=\Big{(}\tilde{\pi},\frac{A^{-1}(\pi,\lambda)\lambda}{|A^{-1}(\pi,\lambda)\lambda|},|A^{-1}(\pi,\lambda)\lambda|A^{-1}(\pi,\lambda)\tau\Big{)}$

is an invertible map and is the natural extension of
$\widetilde{\mathcal{R}}$. Denote by $\widehat{\mu}_{\mathcal{G}}$ the natural
extension of the measure ${\mu}_{\mathcal{G}}$. Then
$\widehat{\mu}_{\mathcal{G}}$ is $\widehat{\mathcal{R}}$-invariant and
$\widehat{\mathcal{R}}$ is recurrent and ergodic with respect to
$\widehat{\mu}_{\mathcal{G}}$.

####  3.1.7. Oseledets splitting

Let us extend the cocycle $A:\mathcal{G}\times\Lambda^{\mathcal{A}}\to
SL_{\mathcal{A}}({\mathbb{Z}})$ to $\widehat{A}:X(\mathcal{G})\to
SL_{\mathcal{A}}({\mathbb{Z}})$ by

$\widehat{A}(\pi,\lambda,\tau):=A(\lambda,\tau)$

and let us consider the cocycle $\widehat{A}:{\mathbb{Z}}\times
X(\mathcal{G})\to SL_{\mathcal{A}}({\mathbb{Z}})$

$\widehat{A}^{(n)}(\pi,\lambda,\tau)=\left\\{\begin{array}[]{cl}\widehat{A}(\pi,\lambda,\tau)\cdot\widehat{A}(\widehat{\mathcal{R}}(\pi,\lambda,\tau))\cdot\ldots\cdot\widehat{A}(\widehat{\mathcal{R}}^{n-1}(\pi,\lambda,\tau))\\!&\\!\text{if
}n\geq 0\\\
\widehat{A}(\widehat{\mathcal{R}}^{-1}(\pi,\lambda,\tau))\cdot\widehat{A}(\widehat{\mathcal{R}}^{-2}(\pi,\lambda,\tau))\cdot\ldots\cdot\widehat{A}(\widehat{\mathcal{R}}^{n}(\pi,\lambda,\tau))\\!&\\!\text{if
}n<0.\end{array}\right.$

Then

(3.8) $\widehat{A}^{(n)}(\pi,\lambda,\tau)={A}^{(n)}(\pi,\lambda)\text{ if
}n\geq 0.$

Let $Y\subset X(\mathcal{G})$ be a subset with
$0<\widehat{\mu}_{\mathcal{G}}(Y)<+\infty$. For a.e. $(\pi,\lambda,\tau)\in Y$
let $r(\pi,\lambda,\tau)\geq 1$ by the first return time of
$(\pi,\lambda,\tau)$ for the map $\widehat{\mathcal{R}}$. Denote by
$\widehat{\mathcal{R}}_{Y}:Y\to Y$ the induced map and by
$\widehat{A}_{Y}:Y\to SL_{\mathcal{A}}({\mathbb{Z}})$ the induced cocycle,
i.e.

$\widehat{\mathcal{R}}_{Y}(\pi,\lambda,\tau)=\widehat{\mathcal{R}}^{r(\pi,\lambda,\tau)}(\pi,\lambda,\tau),\quad\widehat{A}_{Y}(\pi,\lambda,\tau)=\widehat{A}^{(r(\pi,\lambda,\tau))}(\pi,\lambda,\tau)$

for a.e. $(\pi,\lambda,\tau)\in Y$. Let $\widehat{\mu}_{Y}$ be the restriction
of $\widehat{\mu}_{\mathcal{G}}$ to $Y$. Then $\widehat{\mathcal{R}}_{Y}$ is
an ergodic measure-preserving invertible map on $(Y,\widehat{\mu}_{Y})$.

Suppose that $\log\|\widehat{A}_{Y}\|$ and $\log\|\widehat{A}^{-1}_{Y}\|$ are
integrable. Then, by Oseledets theorem, symplecticity of $\widehat{A}_{Y}$
(see [70]) and simplicity of spectrum (see [5]), there exists
$\lambda_{1}>\ldots>\lambda_{g}>0$ such that for a.e. $(\pi,\lambda,\tau)\in
Y$ we have a _Oseledets splitting_

${\mathbb{R}}^{\mathcal{A}}=\bigoplus_{-g\leq i\leq
g}\Gamma_{i}(\pi,\lambda,\tau)$

for which

$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\|\widehat{A}_{Y}^{(n)}(\pi,\lambda,\tau)^{t}v\|=\lambda_{i}$
$\displaystyle\text{ if }v\in\Gamma_{i}(\pi,\lambda,\tau)\text{ and }i>0$
$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\|\widehat{A}_{Y}^{(n)}(\pi,\lambda,\tau)^{t}v\|=-\lambda_{i}$
$\displaystyle\text{ if }v\in\Gamma_{i}(\pi,\lambda,\tau)\text{ and }i<0$
$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\|\widehat{A}_{Y}^{(n)}(\pi,\lambda,\tau)^{t}v\|=0$
$\displaystyle\text{ if }v\in\Gamma_{0}(\pi,\lambda,\tau),$
$\dim\Gamma_{i}(\pi,\lambda,\tau)=1\text{ if }i\neq
0,\quad\dim\Gamma_{0}(\pi,\lambda,\tau)=\kappa-1.$

Furthermore, we have that

$H(\pi)=\bigoplus_{i\neq 0}\Gamma_{i}(\pi,\lambda,\tau).$

We denote by $\Gamma_{s}(\pi,\lambda,\tau)$ and $\Gamma_{u}(\pi,\lambda,\tau)$
the _stable_ and _unstable_ spaces, which are given respectively by

(3.9) $\Gamma_{s}(\pi,\lambda,\tau):=\bigoplus_{-g\leq
i\leq-1}\Gamma_{i}(\pi,\lambda,\tau)\text{ and
}\Gamma_{u}(\pi,\lambda,\tau):=\bigoplus_{1\leq i\leq
g}\Gamma_{i}(\pi,\lambda,\tau).$

Notice that both $\Gamma_{s}(\pi,\lambda,\tau)$ and
$\Gamma_{u}(\pi,\lambda,\tau)$ have exactly dimension $g$. We say in this case
that the Oseledets splitting is _of hyperbolic type_.

####  3.1.8. Veech bases for the kernel $\ker\Omega_{\pi}$

In [64, 65], Veech explicitly defines a bases for $\ker\Omega_{\pi}$ for every
$\pi$ in a given Rauzy class. We recall the construction (which uses the
classical notation for the permutation describing the IETs, also called
_monodromy_ , namely the permutation $\pi_{1}\circ\pi^{-1}_{0}$). Let us first
define the _extended permutation_
$p:\\{0,1,\ldots,d,d+1\\}\to\\{0,1,\ldots,d,d+1\\}$ to be the permutation

$p(j)=\left\\{\begin{matrix}\pi_{1}\circ\pi^{-1}_{0}(j)&\text{ if }&1\leq
j\leq d\\\ j&\text{ if }&j=0,d+1.\end{matrix}\right.$

Following Veech (see [64, 65]), denote by $\sigma=\sigma_{\pi}$ the
corresponding permutation on $\\{0,1,\ldots,d\\}$,

$\sigma(j)=p^{-1}(p(j)+1)-1\text{ for }0\leq j\leq d.$

Notice that (recalling Remark 2.2 and the definition just before of
$\widehat{T}$), we have
$\widehat{T}_{(\pi,\lambda)}r_{\pi_{0}^{-1}(j)}={T}_{(\pi,\lambda)}r_{\pi_{0}^{-1}(\sigma
j)}$ for all $j\neq 0,p^{-1}(d)$.

Denote by $\Sigma(\pi)$ the set of orbits for the permutation $\sigma$. Let
$\Sigma_{0}(\pi)$ stand for the subset of orbits that do not contain zero.
Then $\Sigma(\pi)$ corresponds to the set of singular points of any
translation surface associated to $\pi$ and hence
$\\#\Sigma(\pi)=\kappa(\pi)$.

For every $\mathcal{O}\in\Sigma(\pi)$ denote by
$b(\mathcal{O})\in{\mathbb{R}}^{\mathcal{A}}$ the vector given by

(3.10)
$b(\mathcal{O})_{\alpha}=\chi_{\mathcal{O}}(\pi_{0}(\alpha))-\chi_{\mathcal{O}}(\pi_{0}(\alpha)-1)\text{
for }\alpha\in\mathcal{A},$

where $\chi_{\mathcal{O}}(j)=1$ iff $j\in\mathcal{O}$ and $0$ otherwise.
Moreover, for every $\mathcal{O}\in\Sigma(\pi)$, we denote by

(3.11) $\mathcal{A}^{-}_{\mathcal{O}}=\\{\alpha\in\mathcal{A},\
\pi_{0}(\alpha)\in\mathcal{O}\\},\qquad\mathcal{A}^{+}_{\mathcal{O}}=\\{\alpha\in\mathcal{A},\
\pi_{0}(\alpha)-1\in\mathcal{O}\\}.$

If $\alpha\in\mathcal{A}_{\mathcal{O}}^{+}$ (respectively
$\alpha\in\mathcal{A}_{\mathcal{O}}^{-}$) then the left (respectively right)
endpoint of $I_{\alpha}$ belongs to a separatrix of the saddle represented by
$\mathcal{O}$.

######  Lemma 3.1 (see [65]).

For every irreducible pair $\pi$ we have:

  1. (i)

$\sum_{\mathcal{O}\in\Sigma(\pi)}b(\mathcal{O})=0$;

  2. (ii)

the vectors $b(\mathcal{O})$, $\mathcal{O}\in\Sigma_{0}(\pi)$ are linearly
independent;

  3. (iii)

the linear subspace generated by $\\{b(\mathcal{O})$,
$\mathcal{O}\in\Sigma_{0}(\pi)\\}$ is equal to $\ker\Omega_{\pi}$.

Moreover, $h\in H(\pi)$ if and only if $\langle h,b(\mathcal{O)}\rangle=0$ for
every $\mathcal{O}\in\Sigma(\pi)$.

Veech also describes how these bases change under Rauzy-Veech induction:

######  Lemma 3.2 (see Veech, [65]).

Suppose that
$T_{(\tilde{\pi},\tilde{\lambda})}=\mathcal{R}(T_{(\pi,\lambda)})$. Then there
exists a bijection $\xi:\Sigma(\pi)\to\Sigma(\tilde{\pi})$ such that

$A(\pi,\lambda)^{-1}b(\mathcal{O})=b(\xi\mathcal{O}),\qquad\text{for\ all}\ \
\mathcal{O}\in\Sigma(\pi).$

####  3.1.9. The boundary operator

The following operator $\partial_{\pi}$ is known by _boundary operator_ (as a
special case of the more general operator introduced in [44], see § 4.1.3).
Let $\Sigma(\pi)$ and $\mathcal{A}_{\mathcal{O}}^{\pm}$ be as in the previous
subsection.

######  Definition 1.

Let $\partial_{\pi}:{\mathbb{R}}^{\mathcal{A}}\to{\mathbb{R}}^{\Sigma(\pi)}$
stand for the linear transformation which maps a vector
$h\in{\mathbb{R}}^{\mathcal{A}}$ to the vector in ${\mathbb{R}}^{\Sigma(\pi)}$
whose coordinates $(\partial_{\pi}h)_{\mathcal{O}},\mathcal{O}\in\Sigma(\pi)$
are given by

$(\partial_{\pi}h)_{\mathcal{O}}:=\langle
h,b(\mathcal{O)}\rangle=\sum_{\alpha\in\mathcal{A}_{\mathcal{O}}^{-}}h_{\alpha}-\sum_{\alpha\in\mathcal{A}_{\mathcal{O}}^{+}}h_{\alpha},\qquad\text{for}\
\ \mathcal{O}\in\Sigma(\pi).$

One sees (in light of Remark 2.1) that the image of $\partial_{\pi}$ is:

(3.12)
$\partial_{\pi}({\mathbb{R}}^{\mathcal{A}})=\Big{\\{}(x_{\mathcal{O}})_{\mathcal{O}\in\Sigma(\pi)}:\sum_{\mathcal{O}\in\Sigma(\pi)}x_{\mathcal{O}}=0\Big{\\}}.$

######  Remark 3.3.

We can identify a vector $h\in{\mathbb{R}}^{\mathcal{A}}$ with a piecewise
constant function $g_{h}$, which gives the constant value $h_{\alpha}$ to the
subinterval $I_{\alpha}$. Then the operator $\partial$ can be thought of as
acting on piecewise constant functions and producing, as a value at
$\mathcal{O}\in\Sigma(\pi)$, the _sum of jumps_ of the function $g_{h}$ at the
endpoints corresponding to the singularity labelled by $\mathcal{O}$.

Two extensions of this operator (viewed as in the previous remark as an
operator on functions) will be defined later, to functions piecewise
absolutely continuous on each $I_{\alpha}$ (§ 4.1.3) and to functions with
logarithmic singularities (§ 4.3.3).

####  3.1.10. Boundary operator estimate

Let $H(\pi):=\ker\partial_{\pi}$. Denote by
$p_{H(\pi)}:{\mathbb{R}}^{\mathcal{A}}\to H(\pi)$ the orthogonal projection on
$H(\pi)$ with respect to the standard scalar product on
${\mathbb{R}}^{\mathcal{A}}$.

######  Lemma 3.4.

For any $h\in\mathbb{R}^{\mathcal{A}}$, we have

(3.13) $\|p_{H(\pi)}h\|\leq\sqrt{d}\|h\|.$

Moreover, for any Rauzy class $\mathcal{G}\subset\mathcal{S}^{0}_{A}$ there
exists a positive constant $C_{\mathcal{G}}$ such that for every
$\pi\in\mathcal{G}$ and $h\in{\mathbb{R}}^{\mathcal{A}}$ we have

(3.14) $\|h-p_{H(\pi)}h\|\leq C_{\mathcal{G}}\|\partial_{\pi}h\|.$

###### Proof.

Let $H(\pi)^{\perp}\subset{\mathbb{R}}^{\mathcal{A}}$ be the orthogonal
complement of $H(\pi)$. By Lemma 3.1,
$\partial_{\pi}:H(\pi)^{\perp}\to{\mathbb{R}}^{\Sigma(\pi)}$ is a linear
isomorphism. It follows that there exists $C_{\pi}>0$ such that

$\|h\|\leq C_{\pi}\|\partial_{\pi}h\|\quad\text{ for all }\quad h\in
H(\pi)^{\perp}.$

Hence (3.14) holds with $C_{\mathcal{G}}=\max\\{C_{\pi}:\pi\in\mathcal{G}\\}$.
Denote by $\|\,\cdot\,\|_{2}$ the Euclidean norm on
${\mathbb{R}}^{\mathcal{A}}$. Since $\|h\|_{2}\leq\|h\|\leq\sqrt{d}\|h\|_{2}$
and $p_{H(\pi)}$ is an orthogonal projection, we have

$\|p_{H(\pi)}h\|\leq\sqrt{d}\|p_{H(\pi)}h\|_{2}\leq\sqrt{d}\|h\|_{2}\leq\sqrt{d}\|h\|.$

∎

###  3.2. The Uniform Diophantine-type Condition and its full measure

We will now define the Diophantine-type condition that we will use. First, it
is convenient to introduce an acceleration of Rauzy-Veech induction which
produces times which we call _Rokhlin-balanced_. We then define the condition
and prove that it has full measure.

####  3.2.1. The Rokhlin-balanced acceleration

The following acceleration of Rauzy-Veech induction produces times of the
Rauzy-Veech algorithm where the corresponding Rokhlin towers (see 3.1.4) are
_balanced_ in the sense that all bases have comparable lengths (see (B1) in
Definition 2) and all the towers travel together for a long enough time (see
(B2) in Definition 2). We call these times _Rokhlin-balanced_.

######  Definition 2 (Rokhlin-balance).

Let us say that an accelerating sequence $(n_{k})_{k\geq 0}$ is _Rokhlin-
balanced_ if there exist constants $\kappa>1$ and $0<\delta<1$ such that the
following two conditions hold for every $k\in\mathbb{N}$:

(B1) $\displaystyle|I^{(k)}|\leq\kappa\,|I^{(k)}_{\alpha}|\qquad\text{ for all
}k\geq 1\text{ and }\alpha\in\mathcal{A};$ (B2) for every $k\geq 1$ there
exists a natural number $0<p_{k}\leq\min_{\alpha\in\mathcal{A}}Q_{\alpha}(k)$
such that $\displaystyle\\{T^{i}I^{(k)}:0\leq i<p_{k}\\}\text{ is a Rokhlin
tower of intervals with measure greater than $\delta|I|$.}$

We say that an IET is _Rokhlin-balanced_ if it satisfies Keane’s condition and
it admits a Rokhlin balanced accelerating sequence $(n_{k})_{k\geq 0}$.

######  Remark 3.5.

Notice that by conditions (B1) and (B2), for every $\alpha\in\mathcal{A}$ and
$k\geq 1$ we have

(3.15)
$\displaystyle\|Q(k)\||I^{(k)}|\leq\kappa\sum_{\alpha\in\mathcal{A}}Q_{\alpha}(k)|I^{(k)}_{\alpha}|=\kappa|I|,\text{
and }$ (3.16) $\displaystyle
Q_{\alpha}(k)\lambda^{(k)}_{\alpha}\geq\frac{1}{\kappa}p_{k}|I^{(k)}|\geq\frac{\delta}{\kappa}|I|.$

so that each Rokhlin tower of a balanced acceleration induction time has
measure uniformly bounded below.

Let us show that for almost every IET one can find a Rokhlin-balanced sequence
by considering returns of Rauzy-Veech induction to special compact sets (for
the parameter space of the natural extension, see § 3.1.6). Let us recall that
$X(\mathcal{G})$ denotes the domain of the natural extension of the Rauzy-
Veech induction (see (3.7) in § 3.1.6).

######  Lemma 3.6.

Let $\pi$ be irreducible. For Lebesgue-almost every choice of $\lambda$, the
IET $T=T_{(\pi,\lambda)}$ is Rokhlin-balanced. Furthermore, for every
$0<\delta<1$ one can define a set $Y=Y(\delta)\subset X(\mathcal{G})$ such
that a Rokhlin-balanced accelerating sequence with constant $\delta$ is given
by returns of the natural extension of Rauzy-Veech induction to $Y$.

###### Proof.

Fix $0<\delta<1$. Let us consider a subset $Y=Y(\delta)\subset X(\mathcal{G})$
which satisfies:

  1. (i)

its projection $Y_{0}$ on $\mathcal{G}\times\Lambda^{\mathcal{A}}$ is
precompact with respect to the Hilbert metric;

  2. (ii)

for every $(\pi,\lambda,\tau)\in Y$ we have

$\min\Big{\\{}\Big{\\{}\sum_{\pi_{0}(\alpha)\leq k}\tau_{\alpha}:1\leq
k<d\Big{\\}}\cup\\{h_{\alpha}(\tau):\alpha\in\mathcal{A}\\}\Big{\\}}>\delta\max\\{h_{\alpha}(\tau):\alpha\in\mathcal{A}\\};$

Let $R>0$ be such that
$Y_{0}\subset\mathcal{G}\times\overline{B}_{H}((1/d,\ldots,1/d),R)$, where
$\overline{B}_{H}((1/d,\ldots,1/d),R)$ is the closed ball (with respect to the
Hilbert metric $d_{H}$) of radius $R$ and center at the center of the simplex
$\Lambda^{\mathcal{A}}$.

Balance at visit times. Consider any sequence $(n_{k})_{k\geq 1}$ which
corresponds to visits to the set $Y$. By definition, for every $k$ belonging
to this subsequence, $(\pi^{(k)},\lambda^{(k)},\tau^{(k)})\in Y$. It follows
that $d_{H}\big{(}\lambda^{(k)},(1/d,\ldots,1/d)\big{)}\leq R$. Therefore

$\max_{\alpha\in\mathcal{A}}|I^{(k)}_{\alpha}|/\min_{\alpha\in\mathcal{A}}|I^{(k)}_{\alpha}|\leq
e^{R},$

which implies the condition (B1) for $\kappa:=e^{R}$.

As
$(\pi^{(k)},\lambda^{(k)},\tau^{(k)})=\widehat{\mathcal{R}}^{n_{k}}(\pi,\lambda,\tau))\in
Y$, by condition (ii) in the choice of $Y$, taking

$t^{(k)}:=\min\Big{\\{}\Big{\\{}\sum_{\pi^{(k)}_{0}(\alpha)\leq
l}\tau^{(k)}_{\alpha}:1\leq
l<d\Big{\\}}\cup\\{h^{(k)}_{\alpha}:\alpha\in\mathcal{A}\\}\Big{\\}}\quad(h^{(k)}=h(\tau^{(k)}))$

we have that $I^{(k)}\times[0,t^{(k)}]$ is a rectangle (without singular
points inside) in the translation surface
$M(\pi^{(k)},\lambda^{(k)},\tau^{(k)})$ ($=M(\pi,\lambda,\tau)$) and its area
is greater than

$t^{(k)}\sum_{\alpha\in\mathcal{A}}\lambda^{(k)}_{\alpha}>\delta\max_{\alpha\in\mathcal{A}}h^{(k)}_{\alpha}\sum_{\alpha\in\mathcal{A}}\lambda^{(k)}_{\alpha}\geq\delta\langle\lambda^{(k)},h^{(k)}\rangle=\delta|I|.$

This gives (B2) with
$p_{k}:=[t^{(k)}/\max_{\alpha\in\mathcal{A}}h_{\alpha}(\tau)]$ and
$\delta:=\tfrac{\delta^{2}}{2}<\tfrac{\delta}{2}\tfrac{\min_{\alpha\in\mathcal{A}}h_{\alpha}(\tau)}{\max_{\alpha\in\mathcal{A}}h_{\alpha}(\tau)}$.

Typical Rokhlin balance. It now follows from Poincaré recurrence theorem (and
absolute continuity and finiteness of the Veech invariant measure, see [64])
that almost every IET visits $Y(\delta)$ infinitely often and hence is
Rokhlin-balanced. ∎

####  3.2.2. The Uniform Diophantine Condition definition

The Diophantine-type condition that we will use in the main theorems is the
following.

######  Definition 3 (UDC).

An IET $T:I\to I$ satisfying Keane’s condition, satisfies the _Uniform
Diophantine Condition_ UDC if $T$ is Rokhlin-balanced (in the sense of
Definition 2), and for every $\tau>0$ there exist constants $0<c<C$, a
Rokhlin-balanced accelerating sequence $(n_{k})_{k\geq 0}$ and an increasing
sequence of integers $(r_{n})_{n\geq 0}$ with $r_{0}=0$ and
$r_{n}/n\to\alpha>0$, so that:

(O) $T$ is Oseledets generic, i.e. there exists an extension
$(\pi,\lambda,\tau)$ of $T=T_{(\pi,\lambda)}$ such that it admits an Oseledets
splitting of hyperbolic type, as in § 3.1.7;

and, furthermore, the matrices $Z(k)$ and product matrices $Q(k,l)$ of the
acceleration along the subsequence $(n_{k})_{k\in{\mathbb{N}}}$ (see § 3.1.3)
satisfy the following conditions:

(UDC1) $\displaystyle\|Q_{s}(k,l)\|\leq Ce^{-\lambda(l-k)}\text{ for all
}0\leq k\leq l,\text{ where }\lambda=\lambda_{g}/2;$ (UDC2)
$\displaystyle\|Z(k+1)\|\leq Ce^{\tau|k-r_{n}|}\text{ for all }k\geq 0\text{
and }n\geq 0;$ (UDC3) $\displaystyle ce^{\lambda_{1}k}\leq\|Q(k)\|\leq
Ce^{\lambda_{1}(1+\tau)k}\text{ for all }k\geq 0;$

######  Remark 3.7.

By conditions (UDC2) and (UDC3), there exists $C^{\prime}>0$ such that

(3.17) $\|Z(k+1)\|=O(\|Q(k)\|^{\tau}).$

Then using arguments from Section 1.3.1 in [44], one can show that

(3.18) $\|Q(k)\|=O(\min_{\alpha\in\mathcal{A}}Q_{\alpha}(k)^{1+\tau}).$

Thus, the LABEL:UDC condition implies condition $(a)$ of the Roth-type
Diophantine condition defined in [44]. The other two conditions (as well as
the last assumption of the _restricted_ Roth-type condition202020In [46],
Marmi, Moussa and Yoccoz introduced a more restrictive (but still full
measure) Diophantine-type condition, that they called _restricted Roth-type_ :
in addition to all the properties of _Roth-type_ , one requests in this case
that the stable space has exactly dimension $g$. This holds for IETs which
satisfy the LABEL:UDC in view of the Oseledets genericity assumption (O),
since we require that the splitting is of hyperbolic type, which means exactly
that there are $g$ positive exponents, see § 3.1.7).) also hold, in view of
the Oseledets genericity assumption (O) (see for example Remark 3.4 in [48]).
Thus IETs which satisfy the LABEL:UDC are in particular of (restricted) Roth-
type.

####  3.2.3. Full measure of the LABEL:UDC

Let us show that the LABEL:UDC condition has full measure.

######  Theorem 3.8.

Almost every IET satisfies the LABEL:UDC Diophantine condition.

###### Proof.

We split the proof in several steps.

Construction of a good recurrence set. Let us consider a subset $Y\subset
X(\mathcal{G})$ which satisfies the assumptions (i) and (ii) in the proof of
Lemma 3.6, which guarantees that visits to $Y$ give a Rokhlin-balanced
sequence, and furthermore such that:

  1. (iii)

$\widehat{\mu}(Y)$ is finite, so
$\widehat{\mu}_{Y}:=\widehat{\mu}/\widehat{\mu}(Y)$ is a probability measure;

  2. (iv)

the functions $\log\|\widehat{A}_{Y}\|$ and $\log\|\widehat{A}_{Y}^{-1}\|$ are
integrable with respect to $\widehat{\mu}_{Y}$.

Let $\lambda_{1}>\ldots>\lambda_{g}>0$ the positive Lyapunov exponents of the
corresponding accelerated cocycle, which are $g$ and distinct in view of [22]
and [5]. Let $\lambda:=\lambda_{g}/2$ and $\kappa=de^{R}$. Fix
$0<\tau<\lambda_{g}/2$. Since for $\widehat{\mu}_{Y}$-a.e.
$(\pi,\lambda,\tau)\in Y$ we have

$\displaystyle\lim_{n\to+\infty}\frac{1}{n}\log\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{\Gamma_{s}(\pi,\lambda,\tau)}\|=-\lambda_{g},$

the map from $Y$ to $\mathbb{R}$ given by

$(\pi,\lambda,\tau)\mapsto\sup_{n\geq
0}e^{(\lambda_{g}-\tau)n}\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{\Gamma_{s}(\pi,\lambda,\tau)}\|$

is a.e. defined and measurable. Therefore, there exists a subset $K\subset Y$
with $\widehat{\mu}_{Y}(K)/\widehat{\mu}_{Y}(Y)>1-\tau/2$ and a constant $C>0$
such that if $(\pi,\lambda,\tau)\in K$ then for every $n\geq 0$ we have

(3.19)
$\|\widehat{A}^{(n)}_{Y}(\pi,\lambda,\tau)^{t}\upharpoonright_{\Gamma_{s}(\pi,\lambda,\tau)}\|\leq
Ce^{-(\lambda_{g}-\tau)n}\leq Ce^{-\lambda n}.$

First acceleration. Let us consider the induced map
$\widehat{\mathcal{R}}_{K}:K\to K$ and the induced cocycle
$\widehat{A}_{K}:K\to SL_{\mathcal{A}}({\mathbb{Z}})$. Then
$\widehat{\mathcal{R}}_{K}(\pi,\lambda,\tau)=\widehat{\mathcal{R}}^{r_{K}(\pi,\lambda,\tau)}_{Y}(\pi,\lambda,\tau)$,
where $r_{K}(\pi,\lambda,\tau)\geq 1$ is the first return time of
$(\pi,\lambda,\tau)\in K$ to $K$ for the map $\widehat{\mathcal{R}}_{Y}$. Let
$r_{K}^{(n)}:=\sum_{0\leq i<n}r_{k}\circ\widehat{\mathcal{R}}_{K}$ for every
$n\geq 0$. Then

$\frac{r_{K}^{(n)}}{n}\to\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\text{
a.e.\ on }K$

and furthermore

$\widehat{A}_{K}^{(n)}=\widehat{A}_{Y}^{(r_{K}^{(n)})}\text{ for every }n\geq
0.$

In view of (3.19), for every $(\pi,\lambda,\tau)\in K$ we have

(3.20)
$\|\widehat{A}^{(n)}_{K}(\pi,\lambda,\tau)^{t}\upharpoonright_{\Gamma_{s}(\pi,\lambda,\tau)}\|\leq
Ce^{-\lambda r_{K}^{(n)}(\pi,\lambda,\tau)}\leq Ce^{-\lambda n}$

and for a.e. $(\pi,\lambda,\tau)\in K$ we have

(3.21)
$\lim_{n\to+\infty}\frac{1}{n}\log\|\widehat{A}^{(n)}_{K}(\pi,\lambda,\tau)\|=\lambda_{1}\frac{\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(\lambda_{1},\lambda_{1}(1+\tau)).$

Second acceleration. Since the functions $\log\|\widehat{A}_{K}\|$ and
$\log\|\widehat{A}_{K}^{-1}\|$ are integrable, for a.e. $(\pi,\lambda,\tau)\in
K$ we have
$\log\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{n}(\pi,\lambda,\tau))\|/n\to
0$ as $|n|\to+\infty$, also the map from $K$ to ${\mathbb{R}}$ given by

$(\pi,\lambda,\tau)\mapsto\sup_{n\in{\mathbb{Z}}}e^{-\tau|n|}\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{n}(\pi,\lambda,\tau))\|$

is a.e. defined and measurable. Therefore, there exists a subset
$K^{\prime}\subset K$ with $\widehat{\mu}_{K}(K^{\prime})>0$ and a constant
$C^{\prime}>0$ such that if $(\pi,\lambda,\tau)\in K^{\prime}$ then for every
$n\in{\mathbb{Z}}$ we have

(3.22)
$\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{n}(\pi,\lambda,\tau))\|\leq
C^{\prime}e^{\tau|n|}.$

Moreover, for a.e. $(\pi,\lambda,\tau)\in K^{\prime}$ there exists an
increasing sequence of non-negative integer numbers
$(r_{n}(\pi,\lambda,\tau))_{n\geq 1}$ such that $r_{1}(\pi,\lambda,\tau)=0$
and

(3.23)
$\widehat{\mathcal{R}}_{K}^{r_{n}(\pi,\lambda,\tau)}(\pi,\lambda,\tau)\in
K^{\prime}\text{ for all }n\geq 0\text{ and
}\frac{r_{n}(\pi,\lambda,\tau)}{n}\to\frac{\widehat{\mu}_{K}(K)}{\widehat{\mu}_{K}(K^{\prime})}=:\alpha>0.$

Let $K^{\prime\prime}\subset K^{\prime}$ be a subset of $(\pi,\lambda,\tau)\in
K^{\prime}$ for which (3.21) and (3.23) hold. Then
$\widehat{\mu}_{\mathcal{G}}(K^{\prime\prime})=\widehat{\mu}_{\mathcal{G}}(K^{\prime})>0$.
By the ergodicity of $\widehat{\mathcal{R}}$, for a.e. $(\pi,\lambda,\tau)\in
X(\mathcal{G})$

(3.24) $\text{there exists }n_{1}(\pi,\lambda,\tau)\geq 0\text{ such that
}\widehat{\mathcal{R}}^{n_{1}(\pi,\lambda,\tau)}(\pi,\lambda,\tau)\in
K^{\prime\prime}.$

By Fubini argument, there exists a measurable subset
$\Xi\subset\mathcal{G}\times\Lambda^{\mathcal{A}}$ such that
$\mu_{\mathcal{G}}(\mathcal{G}\times\Lambda^{\mathcal{A}}\setminus\Xi)=0$ and
for every $(\pi,\lambda)\in\Xi$ there exists $\tau\in\Theta_{\pi}$ such that
$(\pi,\lambda,\tau)\in X(\mathcal{G})$ satisfies (3.24).

Full measure. We can now show that every $(\pi,\lambda)\in\Xi$ satisfies the
LABEL:UDC. Suppose that $(\pi,\lambda)\in\Xi$ and $(\pi,\lambda,\tau)\in
X(\mathcal{G})$ satisfies (3.24). Then the corresponding acceleration sequence
$(n_{k})_{k\geq 0}$ is defined by setting $n_{0}:=0$ and then defining $n_{k}$
inductively such that, for every $k\geq 1$,

$\widehat{\mathcal{R}}^{n_{k}}(\pi,\lambda,\tau)=\widehat{\mathcal{R}}_{K}^{k-1}\widehat{\mathcal{R}}^{n_{1}(\pi,\lambda,\tau)}(\pi,\lambda,\tau).$

Let us now consider the cocycle matrices $Z(k)$, $k\in{\mathbb{N}}$, of the
acceleration along the sequence $(n_{k})_{k\in{\mathbb{N}}}$, as defined in §
3.1.3, as well as their products $Q(k,l)$, $k,l\in{\mathbb{N}}$ (see again §
3.1.3). By definition of $Q$ and (3.8), for $1\leq k\leq l$ we have

$\displaystyle Q(k,l)$
$\displaystyle=\widehat{A}_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}$
$\displaystyle Q(0,l)$
$\displaystyle=\widehat{A}_{K}^{(l-1)}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))^{t}\widehat{A}^{(n_{1})}(\pi,\lambda,\tau)^{t}$
$\displaystyle\|Q_{s}(k,l)\|$
$\displaystyle=\|\widehat{A}_{K}^{(l-k)}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}\upharpoonright_{\Gamma_{s}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))}\|$
$\displaystyle\|Q_{s}(0,l)\|$
$\displaystyle\leq\|\widehat{A}_{K}^{(l-1)}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))^{t}\upharpoonright_{\Gamma_{s}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))}\|\|{A}^{(n_{1})}(\pi,\lambda)^{t}\|.$

Since
$\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))\in
K$ for every $k\geq 1$, by (3.20), for $0\leq k<l$ we have

$\|Q_{s}(k,l)\|\leq
Ce^{\lambda}\|{A}^{(n_{1})}(\pi,\lambda)^{t}\|e^{-\lambda(l-k)},$

which gives (UDC1).

Consider now the sequence $(r_{n})_{n\geq 0}$ defined setting $r_{0}:=0$ and,
for $n\geq 1$,

$r_{n}:=r_{n}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))+1.$

As $\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)\in K^{\prime\prime}$, by
(3.23), we have $r_{n}/n\to\alpha>0$ and

$\widehat{\mathcal{R}}_{K}^{r_{n}-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))=\widehat{\mathcal{R}}_{K}^{r_{n}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))}\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)\in
K^{\prime}\text{ for }n\geq 1.$

Since
$Z(k+1)=\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))^{t}$
for $k\geq 1$ and $Z(1)=\widehat{A}^{(n_{1})}(\pi,\lambda,\tau)^{t}$, by
(3.22), for every $n\geq 1$ and $k\geq 1$ we have

$\displaystyle\|Z(k+1)\|$
$\displaystyle=\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{k-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)))\|=\|\widehat{A}_{K}(\widehat{\mathcal{R}}_{K}^{k-r_{n}}(\widehat{\mathcal{R}}_{K}^{r_{n}-1}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))))\|\leq
C^{\prime}e^{\tau|k-r_{n}|}.$

For $k=0$, on the other hand, we have

$\|Z(1)\|=\|{A}^{(n_{1})}(\pi,\lambda)\|\leq\|{A}^{(n_{1})}(\pi,\lambda)\|e^{\tau|r_{n}|}$

for every $n\geq 0$. Moreover, as $r_{1}=1$, it follows that for every $k\geq
1$ we have

$\|Z(k+1)\|\leq C^{\prime}e^{\tau|k-r_{1}|}\leq C^{\prime}e^{\tau|k-r_{0}|},$

which gives (UDC2) with $C=\max(C^{\prime},\|{A}^{(n_{1})}(\pi,\lambda)\|)$.

As $\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau)\in K^{\prime\prime}$, by
(3.21)

$\lim_{k\to+\infty}\frac{\log\|Q(k)\|}{k}=\lim_{k\to+\infty}\frac{\log\|\widehat{A}^{(k-1)}_{K}(\widehat{\mathcal{R}}^{n_{1}}(\pi,\lambda,\tau))\|}{k}=\frac{\lambda_{1}\widehat{\mu}_{Y}(Y)}{\widehat{\mu}_{Y}(K)}\in(\lambda_{1},\lambda_{1}(1+\tau)),$

which implies the condition (UDC3). Finally, the sequence is a Rokhlin-
balanced acceleration sequence by Lemma 3.6, since the set $Y$ was chosen to
satisfies the conditions (i) and (ii) which guarantee Rokhlin-balance in the
proof of Lemma 3.6. This concludes the proof. ∎

###  3.3. Diophantine series

In the proof of our main results, certain sums and series (defined in
Definition 4) which depend on the matrices of the (accelerated) cocycle will
play a central role, both to control Birkhoff sums and to prove ergodicity. We
here show that these quantities, under the LABEL:UDC, are first of all well
defined and furthermore grow in a controlled way (see Proposition 3.9).

######  Definition 4.

For every IET $T:I\to I$ satisfying Keane’s condition and any accelerating
sequence we define four sequences $(K_{l})_{l\geq-1}$,
$(K^{\prime}_{l})_{l\geq-1}$, $(C_{k})_{k\geq 0}$, $(C^{\prime}_{k})_{k\geq
0}$:

$\displaystyle K_{l}(T):=\sum_{j\geq l}\|Z(j+1)\|\|Q_{s}(l,j+1)\|\text{ for
}l\geq 0\text{ and }K_{-1}:=0;$ $\displaystyle K^{\prime}_{l}(T):=\sum_{j\geq
l}\|Z(j+1)\|\|Q_{s}(l,j+1)\|\log\|Q(j)\|\text{ for }l\geq 0\text{ and
}K^{\prime}_{-1}:=0;$ $\displaystyle C_{k}(T):=\sum_{0\leq l\leq
k}\|Q_{s}(l,k)\|\big{(}\|Z(l)\|K_{l-1}(T)+K_{l}(T)\big{)}\text{ for }k\geq 0;$
$\displaystyle C^{\prime}_{k}(T):=\sum_{0\leq l\leq
k}\|Q_{s}(l,k)\|\big{(}\|Z(l)\|K^{\prime}_{l-1}(T)+K^{\prime}_{l}(T)\big{)}\text{
for }k\geq 0.$

Proposition 3.9 below shows in particular that if $T$ satisfies the LABEL:UDC
these quantities are finite and hence well defined for every pairs of integers
$k\geq 0$, $l\geq-1$.

######  Proposition 3.9.

For every IET $T:I\to I$ satisfying the LABEL:UDC all sequences
$(K_{l})_{l\geq-1}$, $(K^{\prime}_{l})_{l\geq-1}$, $(C_{k})_{k\geq 0}$,
$(C^{\prime}_{k})_{k\geq 0}$ are well defined and for every $0<\tau<\lambda/2$
there exists a constant $D>0$ such that

(3.25) $\displaystyle K_{l}(T)$ $\displaystyle\leq De^{\tau(r_{n}-l)}\text{ if
}r_{n-1}\leq l\leq r_{n}\text{ for some }n\geq 0;$ (3.26) $\displaystyle
K^{\prime}_{l}(T)$ $\displaystyle\leq D(l+1)e^{\tau l}\text{ for every }l\geq
0;$ (3.27) $\displaystyle C_{r_{n}}(T)$ $\displaystyle\leq D\text{ for every
}n\geq 1;$ (3.28) $\displaystyle C^{\prime}_{k}(T)$ $\displaystyle\leq
D(k+1)e^{2\tau k}\text{ for every }k\geq 0.$

###### Proof.

By (UDC1) and (UDC2), for $r_{n-1}<l\leq r_{n}$ we have

$\displaystyle K_{l}(T)$ $\displaystyle=\sum_{l+1\leq j\leq
r_{n}}\|Z(j)\|\|Q_{s}(l,j)\|+\sum_{j>r_{n}}\|Z(j)\|\|Q_{s}(l,j)\|$
$\displaystyle\leq C^{2}\sum_{l+1\leq j\leq
r_{n}}e^{\tau(r_{n}-j+1)}e^{-\lambda(j-l)}+C^{2}\sum_{j>r_{n}}e^{\tau(j-1-r_{n})}e^{-\lambda(j-l)}$
$\displaystyle\leq C^{2}e^{\tau(r_{n}-l)}\sum_{j\geq 1}e^{-\lambda
j}+C^{2}e^{-\lambda(r_{n}-l+1)}\sum_{j\geq 0}e^{-(\lambda-\tau)j},$

which gives (3.25).

By condition (UDC3), for all $j\geq l+1$ we have

$\log\|Q(j)\|\leq\log C+\lambda_{1}(1+\tau)j\leq C^{\prime}j\leq
C^{\prime}(l+1)(j-l).$

Therefore, again by (UDC1) and (UDC2), we have

$\displaystyle K^{\prime}_{l}(T)$ $\displaystyle\leq
C^{\prime}(l+1)\sum_{j\geq l+1}\|Z(j)\|\|Q_{s}(l,j)\|(j-l)$ $\displaystyle\leq
C^{\prime}C^{2}(l+1)\sum_{j\geq l+1}(j-l)e^{\tau
j}e^{-\lambda(j-l)}=C^{\prime}C^{2}(l+1)e^{\tau l}\sum_{j\geq
1}je^{-(\lambda-\tau)j},$

which gives (3.26).

In view of (3.25), (UDC1) and (UDC2), we have

$\displaystyle C_{r_{n}}(T)$ $\displaystyle=\sum_{0\leq l\leq
r_{n}}\|Q_{s}(l,r_{n})\|\big{(}\|Z(l)\|K_{l-1}(T)+K_{l}(T)\big{)}$
$\displaystyle\leq C^{2}D\sum_{0\leq l\leq
r_{n}}e^{-\lambda(r_{n}-l)}\big{(}e^{\tau(r_{n}-l+1)}e^{\tau(r_{n}-l+1)}+e^{\tau(r_{n}-l)}\big{)}\leq
2C^{2}De^{2\tau}\sum_{l\geq 0}e^{-(\lambda-2\tau)l},$

which gives (3.27).

In view of (3.26), (UDC1) and (UDC2), for every $k\geq 0$ we have

$\displaystyle C^{\prime}_{k}(T)$ $\displaystyle=\sum_{0\leq l\leq
k}\|Q_{s}(l,k)\|\big{(}\|Z(l)\|K^{\prime}_{l-1}(T)+K^{\prime}_{l}(T)\big{)}$
$\displaystyle\leq C^{2}D\sum_{0\leq l\leq k}e^{-\lambda(k-l)}\big{(}le^{\tau
l}e^{\tau(l-1)}+(l+1)e^{\tau l}\big{)}\leq(k+1)2C^{2}De^{2\tau k}\sum_{j\geq
0}e^{-(\lambda-2\tau)j},$

which gives (3.28). ∎

##  4\. Cocycles with logarithmic singularities

We define in this section norms on the spaces of cocycles
$\varphi:I\to\mathbb{R}$ with logarithmic singularities over IETs that we are
interested in (in view of the reduction explained in § 2.3.3). We first
introduce (in § 4.1) the class of cocycles of bounded variation over a given
IET, then move to cocycles with logarithmic singularities. The norms we
introduce make the space of such cocycles a Banach space. We then prove
several properties which will be used later in the proofs of the main results.

###  4.1. Bounded variation and absolutely continuous cocycles

Let us denote by $\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
the space of functions $\varphi:I\to\mathbb{R}$ such that the restriction
$\varphi:I_{\alpha}\to\mathbb{R}$ is of bounded variation for every
$\alpha\in\mathcal{A}$.

####  4.1.1. Banach structure on bounded variation cocycles

For every function
$\varphi\in\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ and
$x\in I$ we will denote by $\varphi_{+}(x)$ and $\varphi_{-}(x)$ the right-
handed and left-handed limit of $\varphi$ at $x$ respectively. Let us denote
by ${\operatorname{Var}}\left.\left({\varphi}\right)\right|_{J}$ the total
variation of ${\varphi}$ on the interval $J\subset I$. Then set

(4.1)
$\operatorname{Var}\varphi:=\sum_{\alpha\in\mathcal{A}}{\operatorname{Var}}\left.\left(\varphi\right)\right|_{\operatorname{Int}I_{\alpha}}.$

The space $\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ is
equipped with the Banach norm
$\|\varphi\|_{\operatorname{BV}}=\|\varphi\|_{\sup}+\operatorname{Var}\varphi$.

####  4.1.2. Piecewise absolutely continuous cocycles

Denote by $\operatorname{AC}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the
subspace of cocycles in
$\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ which are
absolutely continuous on the interior of each $I_{\alpha}$,
$\alpha\in\mathcal{A}$.

Denote by $\operatorname{BV}^{1}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the
space of functions
$\varphi\in\operatorname{AC}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ such
that
$\varphi^{\prime}\in\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$.
The space $\operatorname{AC}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
equipped with the $\operatorname{BV}$ norm is a Banach space and
$\operatorname{BV}^{1}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ is its dense
subspace.

####  4.1.3. Boundary operator on cocycles

Let
$\partial_{\pi}:\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\to{\mathbb{R}}^{\Sigma(\pi)}$
be the linear operator given by

$(\partial_{\pi}\varphi)_{\mathcal{O}}:=\sum_{\alpha\in\mathcal{A}_{\mathcal{O}}^{-}}\varphi_{-}(r_{\alpha})-\sum_{\alpha\in\mathcal{A}_{\mathcal{O}}^{+}}\varphi_{+}(l_{\alpha})$

for $\mathcal{O}\in\Sigma(\pi)$. This is an extension of the operator defined
in § 3.1.9 from piecewise constant cocycles (in view of Remark 3.3) to bounded
variation cocycles. It associates to each singularity the _sum of jumps_ at
the discontinuities associated to that singularity (see also Remark 3.3).

Remark that if
$\varphi\in\operatorname{AC}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ then

(4.2)
$\sum_{\mathcal{O}\in\Sigma(\pi)}(\partial_{\pi}\varphi)_{\mathcal{O}}=\int_{I}\varphi^{\prime}(x)\,dx=:s(\varphi).$

###  4.2. Cocycles with logarithmic singularities

Consider the space
$\operatorname{LG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ of cocycles with
logarithmic singularities of geometric type on
$\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}$, defined in § 2.3.1 (see in
particular (2.1) for the form of such cocycles), as well as its subspace
$\operatorname{LSG}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$, which consist
of cocycles with logarithmic singularities of geometric type (see § 2.3.1)
satisfying in addition also the symmetry condition (2.2) (both also defined in
§ 2.3.1). We will also use the spaces

$\displaystyle\operatorname{LG}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}):={\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})+{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
$\displaystyle{\operatorname{LSG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}):={\operatorname{LSG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})+{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}),$

consisting of all functions with logarithmic singularities (respectively
symmetric logarithmic singularities) of geometric type of the form (2.1) for
which we require only that
$g_{\varphi}\in\operatorname{BV}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$.
Notice that the space $\operatorname{BV}$ ($\operatorname{AC}$ resp.)
coincides with the subspace of functions
$\varphi\in{\operatorname{LG}}^{\operatorname{BV}}$ ($\operatorname{LG}$
resp.) as in (2.1) such that $C^{\pm}_{\alpha}=0$ for all
$\alpha\in\mathcal{A}$.

####  4.2.1. Norms and Banach space structures

We now define a norm on
${\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
which makes it a Banach space.

######  Definition 5.

For every
$\varphi\in{\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
of the form (2.1) set

$\mathcal{L}(\varphi):=\sum_{\alpha\in\mathcal{A}}(|C^{+}_{\alpha}|+|C^{-}_{\alpha}|),\quad\;\mathcal{LV}(\varphi):=\mathcal{L}(\varphi)+\operatorname{Var}g_{\varphi}.$

The space
${\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
equipped with the norm

$\|\varphi\|_{\mathcal{LV}}=\mathcal{L}(\varphi)+\|g_{\varphi}\|_{\operatorname{BV}}$

becomes a Banach space. Then, since
${\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ and
${\operatorname{LSG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ are closed
subspaces of
${\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
they also inherit the Banach space structure. Moreover, for every
$\varphi\in{\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
we have

(4.3)
$\frac{1}{|I|}\|\varphi\|_{L^{1}(I)}\leq(1+|\log|I||)\|\|\varphi\|_{\mathcal{LV}}.$

Indeed, since every
$\varphi\in{\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
is of the form (2.1), we have

$\frac{1}{|I|}\|\varphi\|_{L^{1}(I)}\leq\frac{\mathcal{L}(\varphi)}{|I|}\int_{I}|\log
x|\,\mathrm{d}x+\frac{\|g_{\varphi}\|_{L^{1}(I)}}{|I|}\leq(1+|\log|I||)\mathcal{L}(\varphi)+\|g_{\varphi}\|_{\sup}.$

We can associate a value also to each _saddle_ in
$\mathrm{Fix}(\psi_{\mathbb{R}})$ individually as follows. Using the notation
introduced in § 3.1.8, let $\mathcal{O}\in\Sigma(\pi)$ be a saddle and let
$\mathcal{A}_{\mathcal{O}}^{-},\mathcal{A}_{\mathcal{O}}^{+}$ be the sets of
letters defined in (3.11), associated respectively to right and left endpoints
of intervals which correspond to this saddle. Then

(4.4)
$\Delta_{\mathcal{O}}(\varphi):=\sum_{\alpha\in\mathcal{A}_{\mathcal{O}}^{-}}C^{-}_{\alpha}-\sum_{\alpha\in\mathcal{A}_{\mathcal{O}}^{+}}C^{+}_{\alpha},$

is the value of the _asymmetry at the saddle_ labelled by $\mathcal{O}$. We
also set

$\mathcal{AS}(\varphi):=\sum_{\mathcal{O}\in\Sigma(\pi)}|\Delta_{\mathcal{O}}(\varphi)|.$

Comparing the above definition and (4.4) with Definition 5, one sees that

(4.5) $\mathcal{AS}(\varphi)\leq\mathcal{L}(\varphi).$

####  4.2.2. Properties of the cocycles arising in the reduction

As we saw in § 2.3, the study of extensions of locally Hamiltonian flows can
be reduced to the study of skew product extensions of IETs with logarithmic
singularities (see Proposition 2.4). We now recall the properties of the
cocycles which appear from this reduction, which were described in [20] (see
the proof of Theorem 6.1 and Proposition 6.1).

Let $M^{\prime}\subset M$ be a minimal component of a locally Hamiltonian flow
$\psi_{\mathbb{R}}$ with non-degenerate saddles. Fix a section $\gamma$ as in
the proof of Proposition 2.4 and consider the map that associate $f\in
C^{2+\epsilon}(M^{\prime})$ to the cocycle $\varphi_{f}$ which appears in the
skew-product presentation of the Poincaré map of the extension
$\Phi^{f}_{\mathbb{R}}$ to $\gamma\times{\mathbb{R}}$ (see Proposition 2.4).

######  Proposition 4.1 (Properties of the skew-products cocycles, see [20]
and in particular212121The statements are all part of Theorem 6.1 in [20], but
$(i)$, namely the boundedness of $\varphi_{|f|}$ when $f\in C^{1}(M)$ and $f$
vanishes on $\mathrm{Fix}(\psi_{\mathbb{R}})\cap M^{\prime}$. This last
statement can be proved with the same arguments used in [20] to prove Theorem
6.1. Theorem 6.1).

For every $\epsilon>0$ the map from $C^{2+\epsilon}(M)$ to
${\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ which maps

$f\mapsto\varphi_{f}\in{\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$

is a bounded linear operator. Moreover,
$g_{\varphi_{f}}^{\prime}\in{\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and there exists $C>0$ such that

$C^{-1}\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap
M^{\prime}}|f(\sigma)|\leq\mathcal{L}(\varphi)\leq
C\sum_{\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap
M^{\prime}}|f(\sigma)|\text{ for every }f\in C^{2+\epsilon}(M).$

Furthermore:

  * (i)

if $f\in C^{1}(M)$ and $f(\sigma)=0$ for all
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})\cap M^{\prime}$ then the map
$\varphi_{|f|}:I\to{\mathbb{R}}$ is bounded;

  * (ii)

If $\psi_{\mathbb{R}}\in\mathcal{U}_{min}$, so $M^{\prime}=M$, then
$\mathcal{AS}(\varphi_{f})=0$ and $\partial_{\pi}(\varphi_{f})=0$.

###  4.3. Properties of cocycles with logarithmic singularities.

We state and prove in this section a number of elementary properties of
cocycles with logarithmic singularities which will be used in the construction
of the correction operators.

####  4.3.1. Control of tails of the derivatives growth

The derivative of a cocycle with logarithmic singularities has singularities
which explode at most as $1/x$, as stated in the following Lemma.

######  Lemma 4.2.

Suppose that ${\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and $g_{\varphi}=0$. For every $\alpha\in\mathcal{A}$ denote by $m_{\alpha}$
the middle point of the interval $I_{\alpha}$, i.e.
$m_{\alpha}:=\tfrac{1}{2}(l_{\alpha}+r_{\alpha})$. Then

(4.6)
$\displaystyle\begin{split}|\varphi^{\prime}(x)(x-l_{\alpha})|&\leq\mathcal{L}(\varphi)\text{
for }x\in(l_{\alpha},m_{\alpha}],\\\
|\varphi^{\prime}(x)(x-r_{\alpha})|&\leq\mathcal{L}(\varphi)\text{ for
}x\in[m_{\alpha},r_{\alpha}).\end{split}$

###### Proof.

Indeed, for every $x\in(l_{\alpha},m_{\alpha}]$ and $\beta\in\mathcal{A}$ we
have

$\Big{\\{}\frac{x-l_{\beta}}{|I|}\Big{\\}}\geq\frac{x-l_{\alpha}}{|I|},\
\Big{\\{}\frac{r_{\beta}-x}{|I|}\Big{\\}}\geq\frac{r_{\alpha}-x}{|I|}\geq\frac{x-l_{\alpha}}{|I|}.$

It follows that

$\displaystyle|\varphi^{\prime}(x)(x-l_{\alpha})|$
$\displaystyle\leq\sum_{\beta\in\mathcal{A}}\frac{|C^{+}_{\beta}|(x-l_{\alpha})}{|I|\\{(x-l_{\beta})/|I|\\}}+\sum_{\beta\in\mathcal{A}}\frac{|C^{-}_{\beta}|(x-l_{\alpha})}{|I|\\{(r_{\beta}-x)/|I|\\}}\leq\sum_{\beta\in\mathcal{A}}(|C^{+}_{\beta}|+|C^{-}_{\beta}|)=\mathcal{L}(\varphi).$

The second inequality of (4.6) follows by the same arguments. ∎

####  4.3.2. Control of mean value on subintervals.

For every integrable function $f:I\to{\mathbb{R}}$ and a subinterval $J\subset
I$ let $m(f,J)$ stand for the _mean value_ of $f$ on $J$, i.e.

$m(f,J)=\frac{1}{|J|}\int_{J}f(x)\,dx.$

######  Proposition 4.3 (Proposition 2.5 in [20]).

If
$\varphi\in{\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$
and $J\subset I_{\alpha}$ for some $\alpha\in\mathcal{A}$, then

(4.7)
$|m(\varphi,J)-m(\varphi,I_{\alpha})|\leq\mathcal{LV}(\varphi)\left(4+\frac{|I_{\alpha}|}{|J|}\right)$

and

(4.8) $\frac{1}{|J|}\int_{J}|\varphi(x)-m(\varphi,J)|\,dx\leq
8\mathcal{LV}(\varphi).$

######  Lemma 4.4.

Let
$\varphi\in{\operatorname{LG}}^{\operatorname{BV}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$.
Then for every $x\in\operatorname{Int}I_{\alpha}$ we have

(4.9) $\displaystyle\begin{aligned}
|\varphi(x)-m(\varphi,[l_{\alpha},m_{\alpha}])|&\leq\mathcal{LV}(\varphi)\Big{(}1+\log\frac{|I_{\alpha}|}{(x-l_{\alpha})}\Big{)}\text{
if }x\in(l_{\alpha},m_{\alpha}],\\\
|\varphi(x)-m(\varphi,[m_{\alpha},r_{\alpha}])|&\leq\mathcal{LV}(\varphi)\Big{(}1+\log\frac{|I_{\alpha}|}{(r_{\alpha}-x)}\Big{)}\text{
if }x\in[m_{\alpha},r_{\alpha}).\end{aligned}$

###### Proof.

Step 1: First note that for any $C^{1}$-map $f:(x_{0},x_{1}]\to{\mathbb{R}}$
such that $|f^{\prime}(x)(x-x_{0})|\leq C$ for $x\in(x_{0},x_{1}]$, we have
that for all $t,s\in(x_{0},x_{1}]$

$|f(s)-f(t)|=|\int_{t}^{s}f^{\prime}(u)\,du|\leq
C|\int_{t}^{s}\frac{1}{u-x_{0}}\,du|=C|\log\frac{t-x_{0}}{s-x_{0}}|$

and hence that

(4.10)
$\displaystyle\begin{split}|f(s)-m(f,[x_{0},x_{1}])|&\leq\frac{C}{x_{1}-x_{0}}\int_{x_{0}}^{x_{1}}|\log\frac{t-x_{0}}{s-x_{0}}|\,dt\\\
&=C\Big{(}\log\frac{x_{1}-x_{0}}{s-x_{0}}+1-2\frac{x_{1}-s}{x_{1}-x_{0}}\Big{)}\leq
C\Big{(}\log\frac{x_{1}-x_{0}}{s-x_{0}}+1\Big{)}.\end{split}$

Step 2: Suppose now that
$\varphi\in{\operatorname{LG}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ with