# Analytic reparametrizations of translation toral flows with countable
Lebesgue spectrum

Fatna Abdedou, Bassam Fayad, Arezki Kessi

###### Abstract

We give an example of a real analytic reparametrization of a minimal
translation flow on $\mathbb{T}^{5}$ that has a Lebesgue spectrum with
infinite multiplicity.

##  1 Introduction

Inspired by Kolmogorov’s 1954 ICM talk [8], the following questions arise: Can
a completely integrable Hamiltonian flow be perturbed in the real analytic
category so that the perturbed flow has an invariant torus exhibiting dynamics
with a maximal spectral type equivalent to the Lebesgue measure on
$\mathbb{R}$? Can a reparametrized translation flow of the torus have a
maximal spectral type that is equivalent to the Lebesgue measure on
$\mathbb{R}$?

By a classical construction that we recall below, a positive answer to the
second question, with reparametrization functions arbitrarily close to 1,
immediately yields examples of perturbations that answer positively the first
question. The aim of this paper is to prove the following. Let
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$.

######  Theorem 1.

There exists a minimal translation flow on $\mathbb{T}^{5}$, and a strictly
positive real entire function $\Phi$ defined on $\mathbb{T}^{5}$, such that
the reparametrization of the irrational flow $(\alpha,1)$ by $\Phi$ has a
Lebesgue spectrum with infinite multiplicity. Moreover, $\Phi$ can be chosen
arbitrarily close to $1$ on any bounded domain.

Note that arbitrarily small perturbations of the reparametrized flow, such as
changing slightly $\alpha$ or $\Phi$, can render the flow analytically
conjugated to a translation flow, who has a pure point spectrum.

Since the reparametrization function can be chosen arbitrarily close to $1$,
it is a classical observation that this automatically gives examples of
perturbations of integrable Hamiltonians with invariant tori carrying uniquely
ergodic dynamics with infinite Lebesgue spectrum (ILS). Moreover, the
integrable Hamiltonian can be chosen to be convex. Indeed, consider the
completely integrable system on $\mathbb{T}^{5}\times\mathbb{R}^{5}$ given by

$H(\theta,r)=\Phi(\theta)(\frac{1}{2}\sum_{j=1}^{5}r_{j}^{2}-1).$

Denote $X^{t}_{H}$ the corresponding flow. The energy surface $\\{H=0\\}$ is
foliated by invariant tori for $X^{t}_{H}$ on which the restricted flow is a
reparametrization by $\Phi$ of the translation flow of frequency vector $r$.
One can then pick the vector $r$ and the function $\Phi$ to be as in the
theorem and get the invariant torus with ILS.

Kolmogorov [7] showed that reparametrizations of translation flows on
$\mathbb{T}^{2}$ are typically conjugated to translation flows, since it
suffices for this that the slope of the translation flow belongs to the full
measure set of Diophantine numbers. He also observed that more exotic
behaviors should be expected for the reparametrized flows in the case of
Liouville slopes. Shklover indeed obtained in [shklover] examples of real
analytic reparametrizations of linear flows on the $2$-torus that were weak
mixing (continuous spectrum). Not long after Shklover’s result, Katok [4] and
Kochergin [5] showed the absence of mixing for non-singular conservative
$C^{1}$ flows on the $2$-torus. Such flows can be viewed as special flows
above a circular rotation $R_{\alpha}$ and the absence of mixing is based on
the Denjoy-Koksma cancellation property (DKP) for Birkhoff sums above an
irrational rotation. In the case of a special flow with a $C^{1}$ ceiling
function of average $1$, one can easily see from the DKP that the special flow
satisfies $T^{q_{n}}\to{\rm Id}$ uniformly, where $q_{n}$ is the sequence of
denominators of the best rational approximations (or convergents) of $\alpha$.

For reparametrizations of minimal translation flows on higher dimensional tori
the situation is quite different. Kolmogorov’s linearization result still
holds for almost all frequency vectors as observed by Herman. However, Yoccoz
showed in [9] that the Denjoy-Koksma cancellation property has no counterpart
in higher dimensions. Using frequency vectors $\alpha\in\mathbb{R}^{2}$ with
coordinates that are Liouville numbers with a special alternating
configuration of the denominators of their convergents, he constructed a
function of which the Birkhoff sums above the (minimal) translation
$R_{\alpha}$ do not have the DKP.

More precisely, the main ingredient in the construction of [9] is the use of a
vector $\alpha=(\alpha_{1},\alpha_{2})$ such that the denominators,
$q_{n}^{(1)}$ and $q_{n}^{(2)}$ of the convergents of $\alpha_{1}$ and
$\alpha_{2}$ are alternated and such that each term in the increasing sequence
$\ldots,q_{n}^{(1)},q_{n}^{(2)},q_{n+1}^{(1)},q_{n+1}^{(2)},\ldots$ is
exponentially larger than the precedent one. Then [9] constructs a real
analytic function $A:\mathbb{T}^{2}\to\mathbb{C}$ with zero integral such that
for almost every $(x_{1},x_{2})\in\mathbb{T}^{2}$,
$|A_{m}(x_{1},x_{2})|\to\infty$ as $m\to\infty$.

Using the construction of Yoccoz, the second author constructed in [1] mixing
analytic reparametrizations for a class of minimal translation flows on
$\mathbb{T}^{3}$. These flows can be viewed as special flows above a minimal
translations $R_{\alpha}$ of the two torus with $\alpha$ as in [9], and the
mixing mechanism comes from the uniform stretch of the Birkhoff sums of an
adequately chosen ceiling function $\varphi$ above $R_{\alpha}$ (see Figure
Section 2 and Section 2 for the definition of the uniform stretch measurement
$S_{J}^{t}$).

To be more precise, the disposition of the best approximations of $\alpha_{1}$
and $\alpha_{2}$, allows to consider a function $\varphi$ that is a sum of two
functions $\varphi_{1}(x_{1})+\varphi_{2}(x_{2})$ such that the ergodic sums
$\varphi_{m}$ of the function $\varphi$, for any $m$ sufficiently large, will
be always stretching (i.e. have big derivatives at most points), in one or in
the other of the two directions, $x_{1}$ or $x_{2}$, depending on whether $m$
is far from $\\{q_{n}\\}$ or far from $\\{q^{\prime}_{n}\\}$. And this stretch
will increase when $m$ goes to infinity. So when time goes from $0$ to $t$,
$t$ large, the image of a small typical interval $J$ from the basis
${\mathbb{T}}^{2}$ (depending on $t$ the intervals should be taken along the
$x_{1}$ or the $x_{2}$ axis) will be more and more stretched in the fibers’
direction, until the image of $J$ at time $t$ will consist of a lot of almost
vertical curves whose projection on the basis lies along a piece of a
trajectory under the translation $R_{\alpha}$. By unique ergodicity these
projections become more and more uniformly distributed, and so will
$T_{\alpha,\varphi}^{t}(J)$. For each $t$, and except for increasingly small
subsets of it (as function of $t$), it is possible to cover the basis with
such ”good” intervals. Besides, what is true for $J$ on the basis is true for
$T_{\alpha,\varphi}^{s}(J)$ at any bounded height $s$ on the fibers. So
applying Fubini Theorem, we will obtain the asymptotic uniform distribution of
any measurable subset, which is, by definition, the mixing property (see
Figure 1).

Figure 1: Mixing mechanism for special flows: the image of a rectangle is a
union of long narrow strips which fill densely the phase space.

Uniform stretch is also responsible for mixing in the conservative surface
flows with one degenerate singularity studied by Kochergin in the 1970s [6].
Kochergin flows are special flows under an integrable ceiling function with at
least one power singularity (see Figures 2 and 3 and the precise definition of
special flows in Section 2). The uniform stretch of the Birkhoff sums in this
context comes from the shear between different orbits as they go near the
singularity.

Figure 2: Representation of a $2$-torus flow with one degenerate saddle as a
special flow under a ceiling function with a power-like singularity.

The paper [2] proved ILS for Kochergin flows. To start with, it established
square summable decay of correlations for observables that are smooth
coboundaries above the flow. This suffices to conclude that the maximal
spectral type is absolutely continuous with respect to the Lebesgue measure.
The use of smooth coboundaries to get faster decay estimates than for general
smooth observables or characteristic functions was inspired from [3] that
established Lebesgue maximal spectral type for time changes of horocyclic
flows. The crucial point in the use of smooth coboundaries is that it allows
to translate all the uniform stretch in speed of mixing.

To establish ILS for Kochergin flows, as well as for time changes of
horocyclic flows, the paper [2] introduced a criterion based on the decay of
correlations that allows to deduce the ILS property. The criterion exploits
the speed of equi-distribution of small sets transversal to the flow direction
to build observables that have almost orthogonal cyclic spaces and that have a
spectral type close to being Lebesgue. An abstract statement then allows to
conclude the ILS property.

The delicate point with Kochergin flows, in comparison with time changes of
horocyclic flows for example, is that the shear between orbits is not uniform.
Equivalently, the stretch of the Birkhoff sums of the ceiling function can
vanish or be very weak on parts of the phase space in a way that is also not
uniform in time. The key factor is how close to the singularity different
orbits get to before time $t$.

In the mixing reparametrizations of translation flows the same phenomenon of
non uniform shear in space and time appears because the DKP still appears for
some special times on part of the phase space. A precise computation shows
that, at time $t$, given a direction of uniform stretch, the set with uniform
stretch weaker than $t^{1-2\beta}$ is of measure that can be compared to
$t^{-\beta+\epsilon}$, for any $\epsilon>0$ (see Corollary 3). This means that
for times with only one direction of stretch as it happens for the special
flows above $\mathbb{T}^{2}$, we will get a uniform stretch larger than
$t^{1/2}$ away from a set of measure $t^{-1/4+\epsilon}$. The size of the bad
set in this case is too large since we seek a square summable decay of
correlations. However, if for each $t$ there are 3 independent directions of
stretch, the stretch will be stronger than $t^{1/2+\epsilon}$ on a set of
measure $t^{-3/4+3\epsilon}=o(t^{-1/2-\epsilon})$.

A simple observation concerning the construction of the frequency vectors as
in [9, 1] allows to construct frequencies $\alpha\in\mathbb{R}^{4}$ and
ceiling functions $\varphi$ such that for each time $t$ there are three
directions of uniform stretch and gives examples of special flows above
$\mathbb{T}^{4}$ translations with ILS. This corresponds to reparametrizations
of minimal flows on $\mathbb{T}^{5}$. Our method, that insures that for each
$t$ the decay is less than $t^{-1/2-\epsilon}$, does not allow to treat the
case of $\mathbb{T}^{3}$ and $\mathbb{T}^{4}$.

In principle one could apply the strategy adopted in [2] to the case of the
mixing reparametrizations on $\mathbb{T}^{3}$ or $\mathbb{T}^{4}$. That is,
accept the existence of a sequence of special times $(t_{n})$ (such as the
multiples of the denominators of the convergents) at which the small measure
sets that are bad (with no strong uniform stretch) lead to a decay slower than
$t_{n}^{-1/2}$, but still recover square summable correlations due to the fact
that for most of the times that are in a medium scale neighborhood of the
times $t_{n}$, there is some small power decay of correlations on the bad set
itself. This is what was done for Kochergin flows in [2] because the decay was
slower than $t^{-1/2}$ for some special times. We believe the same can be done
for reparametrized mixing flows on $\mathbb{T}^{4}$ and possibly on
$\mathbb{T}^{3}$, but the proof would then be much more technical than the one
given here for reparametrizations on $\mathbb{T}^{5}$.

##  2 Notations and definitions

  * •

Special flows above translations of the torus. Let
$R_{\alpha}:\mathbb{T}^{d}\to\mathbb{T}^{d}$,
$R_{\alpha}(\theta)=\theta+\alpha\;\text{\rm mod }1$, where
$\alpha\in[0,1]^{d}$ is a vector such that $1,\alpha_{1},\ldots,\alpha_{d}$
are independent over $\mathbb{Z}$. Let $\varphi\in
L^{1}(\mathbb{T}^{d},\lambda_{\mathbb{T}^{d}})$ be a strictly positive
function. We recall that the special flow $T^{t}:=T^{t}_{\alpha,{\varphi}}$
constructed above $R_{\alpha}$ and under $\varphi$ is the flow defined almost
everywhere by

$\displaystyle\mathbb{T}^{d}\times\mathbb{R}/\sim$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{T}^{d}\times\mathbb{R}/\sim$ $\displaystyle(\theta,s)$
$\displaystyle\rightarrow$ $\displaystyle(\theta,s+t),$

where $\sim$ is the identification, defined on
$\mathbb{T}^{d}\times\mathbb{R}$,

$(\theta,s+\varphi(\theta))\sim(R_{\alpha}(\theta),s)\,.$

Equivalently (see Figure 3), this special flow is defined for $s\geq 0$ and
for all times $t\in\mathbb{R}$ such that $t+s\geq 0$ (with a similar
definition for times $t\in\mathbb{R}$ such that $t+s<0$) by

$T^{t}(\theta,s)=(\theta+N(\theta,s,t)\alpha,t+s-S_{N(\theta,s,t)}\varphi(\theta)),$
(2 .1)

where $N(\theta,s,t)$ is the unique integer such that

$0\leq
t+s-S_{N(\theta,s,t)}{\varphi}(\theta)\leq{\varphi}(\theta+N(\theta,s,t)\alpha),$
(2 .2)

and

$S_{n}{\varphi}(\theta)=\left\\{\begin{array}[]{ccc}{\varphi}(\theta)+\ldots+{\varphi}(R_{\alpha}^{n-1}\theta)&\mbox{if}&n>0\\\
0&\mbox{if}&n=0\\\
-({\varphi}(R_{\alpha}^{n}\theta)+\ldots+{\varphi}(R_{\alpha}^{-1}\theta))&\mbox{if}&n<0.\end{array}\right.$

Figure 3: The orbit of a point by the special flow above a transformation $f$
and under a bounded ceiling function $\tau$.

  * •

We recall the notations $M=\\{(x,s)\in\mathbb{T}^{4}\times\mathbb{R}\;:\;0\leq
s<\varphi(x)\\}$ for the configuration space of the flow
$T^{t}_{\alpha,\varphi}$ and $\mu$ for the measure equal to the restriction to
$M$ of the product of the Haar measures
$\lambda_{4}:=\lambda_{\mathbb{T}^{4}}$ on the torus $\mathbb{T}^{4}$ and
$\lambda:=\lambda_{\mathbb{R}}$ on the real line $\mathbb{R}$.

For a given $\zeta>0$, let us denote

$M_{\zeta}:=\\{(x,s)\in M\;:\;0\leq s\leq\varphi(x)-\zeta\\}.$ (2 .3)

  * •

For $j\in\\{0,\ldots,4\\}$, we denote by $I^{j}$ intervals in the
$j$-direction : $I^{j}\subset M$ is of the form
$\\{z=(x_{1},x_{2},x_{3},x_{4},s)\in M:x_{j}\in[a,b]\\}$, for some $0\leq
a<b\leq 1$. We also call such intervals $j$-intervals.

  * •

For a point $z\in M$, we denote by $\bar{z}$ its projection on the base
$\mathbb{T}^{4}$. We denote by $\pi_{j}(z)$ the projection of $z$ on the
$j^{\rm th}$-coordinate of $\mathbb{T}^{4}$.

  * •

On a $j$-interval $I^{j}$, we denote by $\lambda$ the Lebesgue measure.

  * •

Given an interval $I\subset M$ (in any direction) and $t\in\mathbb{R}$, we
define

$\displaystyle r^{t}_{I}$ $\displaystyle=\inf_{z\in
I}\lvert\varphi^{\prime}_{N(z,t)}(\bar{z})\rvert$ (2 .4) $\displaystyle
S^{t}_{I}$ $\displaystyle=\inf_{x\in
I}\frac{(\varphi^{\prime}_{N(z,t)}(\bar{z}))^{2}}{\varphi^{\prime\prime}_{N(z,t)}(\bar{z})}.$
(2 .5)

The quantity $S^{t}_{I}$ measures the uniform stretch of the Birkhoff sums
above the interval $I$. Indeed, when $S^{t}_{I}$ is large, this corresponds to
almost linear expansion of the interval $I\times\\{0\\}\subset M$ under the
flow at time $t$.

##  3 The construction and precise statements

###  3.1 The choice of the frequency vector

Following [9] (see also [1] in the context of special flows), let
$Y\subset[0,1]^{4}$ be the set of vectors
$\alpha:=(\alpha_{1},...,\alpha_{4})$ whose sequences of best approximations
$q_{n}^{(1)}$,…, $q_{n}^{(4)}$ satisfy the following for all $n\geq 1$ :

$q_{n}^{(2)}\geq e^{nq_{n}^{(1)}},\quad q_{n}^{(3)}\geq e^{nq_{n}^{(2)}},\quad
q_{n}^{(4)}\geq e^{nq_{n}^{(3)}},\quad q_{n+1}^{(1)}\geq e^{nq_{n}^{(4)}}.$ (3
.1)

As in [9], we have that

######  Lemma 1.

The set $Y$ is a dense continuum in $[0,1]^{4}$.

###### Proof of Lemma 1.

First recall that any irrational number $\alpha\in\mathbb{R-Q}$ can be written
as a continued fraction expansion

$\alpha=\left[a_{0},a_{1},...\right]=a_{0}+1/(a_{1}+1/(a_{2}+...1/a_{n-1}+1/a_{n})...),$

where $\left\\{a_{j}\right\\}_{j\geq 1}$ is a sequence of integers $\geq 1$
and $a_{0}=\left[\alpha\right].$ Conversely, any infinite sequence
$\left\\{a_{j}\right\\}_{j\geq 1}$ corresponds to a unique number $\alpha$.
The convergents
$\left(p_{n},q_{n}\right)\in\mathbb{Z}\times\mathbb{Z}^{\ast}$of $\alpha$ are
defined by $a_{j}$ in the following way

$\left\\{\begin{array}[]{ccc}p_{n}=a_{n}p_{n-1}+p_{n-2}&for\text{ }n\geq
2,\text{ }&p_{0}=a_{0},\text{ }p_{1}=a_{0}a_{1}+1\\\
q_{n}=a_{n}q_{n-1}+q_{n-2}&for\text{ }n\geq 2,&q_{0}=1,\text{
}q_{1}=a_{1}.\end{array}\right.$ (3 .2)

For an arbitrary choice $n_{0}$ and an arbitrary choice of $a_{n}^{(j)}$, for
$j\in\\{0,\ldots,4\\},n\leq n_{0}$, it is straightforward from (3 .2) to
construct inductively
$a_{n_{0}+1}^{(1)},a_{n_{0}+1}^{(2)},a_{n_{0}+1}^{(3)},a_{n_{0}+1}^{(4)},a_{n_{0}+2}^{(1)},\ldots$
such that (3 .1) holds. Indeed, suppose all coefficients are chosen up to
$a_{n}^{(4)}$. Then we just have to take $a_{n+1}^{(1)}>e^{nq_{n}^{(4)}}$,
then $a_{n+1}^{(2)}>e^{(n+1)q_{n+1}^{(1)}}$, then
$a_{n+1}^{(3)}>e^{(n+1)q_{n+1}^{(2)}}$, then
$a_{n+1}^{(4)}>e^{(n+1)q_{n+1}^{(3)}}$. Since each time we have to pick a
coefficient we can choose it in an infinite semi-interval of the integers, the
set $Y$ is clearly a continuum. The fact that the coefficients can be chosen
arbitrarily for $n\leq n_{0}$ with $n_{0}$ arbitrarily large, implies the
density of the set $Y$. ∎

Define for $j\in\\{0,\ldots,4\\}$

$T_{n}^{j}=[e^{nq_{n}^{(j)}},\frac{q_{n+1}^{(j)}}{n+1}]$ (3 .3)

The important consequence from our definition of the set of $Y$ is the
following

######  Lemma 2.

Every $t\geq 0$ belongs to at least three intervals $T_{n_{j}}^{j}$. More
precisely one of the following holds

  1. 1.

$t\in T_{n}^{1}\bigcap T_{n}^{2}\bigcap T_{n}^{3}$

  2. 2.

$t\in T_{n}^{2}\bigcap T_{n}^{3}\bigcap T_{n}^{4}$

  3. 3.

$t\in T_{n}^{3}\bigcap T_{n}^{4}\bigcap T_{n+1}^{1}$

  4. 4.

$t\in T_{n}^{4}\bigcap T_{n+1}^{1}\bigcap T_{n+1}^{2}$

###### Proof of Lemma 2.

Let $n$ be such that $t\in[e^{nq_{n}^{(1)}},e^{(n+1)q_{n+1}^{(1)}}]$. If
$t\leq q_{n}^{(2)}/n$, then $t\in T_{n-1}^{j}$ for every $j=2,3,4$. If
$t\in[q_{n}^{(2)}/n,q_{n}^{(3)}/n]$, then $t\in T_{n-1}^{j}$, for $j=3,4$ and
$t\in T_{n}^{(1)}$. If $t\in[q_{n}^{(3)}/n,q_{n}^{(4)}/n]$, then $t\in
T_{n-1}^{4}$, $t\in T_{n}^{(j)}$, for $j=1,2$. ∎

###  3.2 The choice of the ceiling function

Following [9, 1], let $\varphi$ be the following strictly positive real
analytic function on $\mathbb{T}^{4}$

$\varphi(x_{1},...,x_{4})=1+\sum_{j=1}^{4}\sum_{n\geq n_{0}}\frac{\cos(2\pi
q^{(j)}_{n}x_{j})}{e^{q^{(j)}_{n}}},$

where $n_{0}$ is chosen sufficiently large so that $\varphi$ is strictly
positive.

###  3.3 The special flows

Now, for $\alpha\in Y$ and the function $\varphi$, we denote by
$\\{T^{t}_{\alpha,\varphi}\\}$ the special flow above the translation
$R_{\alpha}$ on $\mathbb{T}^{4}$ and under the ceiling function $\varphi$ (see
Section 2 for the definitions). Our main result is the following

######  Theorem 2.

For any $\alpha\in Y$, the special flow constructed over the translation
$R_{\alpha}$ on $\mathbb{T}^{4}$ and under the ceiling function $\varphi$ has
countable Lebesgue spectrum.

From Theorem 2 and the correspondence between special flows above translations
and reparametrizations of translation flows on the torus, we derive the
following result that gives a precise example of a flow as in Theorem 1.

######  Corollary 1.

For any $\alpha\in Y$, there exists a strictly positive real analytic function
$\Phi$ defined on $\mathbb{T}^{5}$, such that the reparametrization of the
irrational flow $(\alpha,1)$ by $1/\Phi$ has countable Lebesgue spectrum.

###### Proof.

We sketch the proof and refer to [1, Proposition 6] for the details. Define

$\Phi(x_{1},\ldots,x_{5})=1+\Re\left(\sum_{n=n_{0}}^{\infty}d^{(j)}_{n}e^{i2\pi(q_{n}^{(j)}x_{j}+l_{n}^{(j)}x_{5})}\right)$

where we choose $l_{n}^{(j)}$ to be the closest relative integer to
$-q_{n}^{(j)}\alpha_{j}$, and

$d^{(j)}_{n}=\frac{i2\pi(q_{n}^{(j)}\alpha_{j}+l_{n}^{(j)})}{e^{i2\pi(q_{n}^{(j)}\alpha_{j}+l_{n}^{(j)})}-1}e^{-q_{n}^{(j)}}.$

Then, if $n_{0}$ is sufficiently large, $\Phi$ is a real analytic strictly
positive function on $\mathbb{T}^{5}$, that satisfies

$\varphi(x_{1},\ldots,x_{4})=\int_{0}^{1}\Phi(x_{1}+s\alpha_{1},\ldots,x_{4}+s\alpha_{4},s)ds.$

Hence the reparametrization of the flow of frequency $(\alpha,1)$ with the
function $1/\Phi$ can be viewed as the special flow above $R_{\alpha}$ on
$\mathbb{T}^{4}$ with the ceiling function $\varphi$. Hence the function
$\Phi$ satisfies the conclusion of Corollary 1.∎

###  3.4 Square summable decay for smooth coboundaries

We recall that $f$ is called a smooth coboundary over the flow
$T^{t}_{\alpha,\varphi}$ if there exists a smooth function $\phi$ such that,
for any $a<b$,

$\int_{a}^{b}f(T^{t}(x_{0},t_{0}))dt=\int_{a}^{b}f(x_{0},t_{0}+t)dt=\phi(x_{0},t_{0}+b)-\phi(x_{0},t_{0}+a).$

The function $\phi$ is called the transfer function of $f$. The space of
smooth coboundaries is dense in the subspace $L^{2}_{0}(M,\mu)\subset
L^{2}(M,\mu)$ of zero average functions, provided $T^{t}_{\alpha,\varphi}$ is
ergodic (which is always the case if $\alpha$ is irrational). The set of
smooth functions that are supported inside $M_{\zeta}$ for some $\zeta>0$ is
also dense in the subspace $L^{2}_{0}(M,\mu)\subset L^{2}(M,\mu)$ of zero
average functions.

Hence to prove that the maximal spectral type of $T^{t}_{\alpha,\varphi}$ is
absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$, it
suffices to show the following

######  Theorem 3.

For every $\zeta>0$, for every smooth coboundary function $f$ and every smooth
function $g$ supported in $M_{\zeta}$, there exists a constant
$C(\alpha,\varphi,\zeta)>0$ such that, for every $t$, it holds that

$\left\lvert\int_{M}f(T^{t}_{\alpha,\varphi}z)g(z)d\mu\right\rvert\leq
C\mathcal{N}_{1}(f,g)t^{-1/2-\epsilon}.$

The symbol $\mathcal{N}_{1}(f,g)$ will denote positive constant depending on
the $C^{0}$ and the $C^{1}$ norm of the functions $f$ and $g$ (see Section 5
for the definitions).

###  3.5 Lebesgue spectrum with infinite multiplicity

To show that the spectral type is Lebesgue and has infinite multiplicity we
apply a general criterion, based on correlation decay, similar to the one used
in [2]. We state a version of the criterion adapted to our context in Section
6 and then we show that it holds for the flow $T^{t}_{\alpha,\varphi}$ defined
in Section 3.3.

##  4 Birkhoff sums and their uniform stretch on good intervals

First of all we state a standard result on the uniform behavior of $N(x,t)$
due to the unique ergodicity of $R_{\alpha}$ and continuity of $\varphi$.

######  Lemma 3.

For any $t$ sufficiently large, for any $x\in\mathbb{T}^{4}$

$N(x,t)\in[\frac{t}{2},2t].$

###### Proof.

By the definition of $N(x,t)$,

$0\leq t-\varphi_{N(x,t)}(x)\leq\varphi(R^{N(x,t)}_{\alpha}x)\leq\|\varphi\|.$

This shows that $N(x,t)\to\infty$ uniformly as $t\to\infty$. Since by unique
ergodicity of $R_{\alpha}$ we have that
$\varphi_{N(x,t)}/N(x,t)\to\int_{\mathbb{T}^{4}}\varphi(x)dx=1$, we get the
bounds of the lemma. ∎

For $n\in\mathbb{N}$ and $\theta>0$ we define for every $j\in\\{1,\ldots,4\\}$

$\mathcal{W}(n,\theta,j)=\left\\{x\in\mathbb{T}^{1}/\\{q_{n}^{(j)}x\\}\in[\theta,\frac{1}{2}-\theta]\cup[\frac{1}{2}+\theta,1-\theta]\right\\}$

######  Definition 1 (Good intervals).

We say that a $j$-interval $I^{j}\subset M$ is $(n,\theta,j)$-good if
$\pi_{j}(I^{j})\bigcap\mathcal{W}(n,\theta,j)=\emptyset$.

Recall the definition of
$T_{n}^{j}=[e^{nq_{n}^{(j)}},\frac{q_{n+1}^{(j)}}{n+1}]$ given in (3 .3). Let

${\bar{T}}_{n}^{j}:=[e^{nq_{n}^{(j)}}/10,10\frac{q_{n+1}^{(j)}}{n+1}],\quad
j=1,...,4.$

The following estimate on uniform stretch is derived from a similar one in [1]
where it plays the key role for mixing of the reparametrized flows on
$\mathbb{T}^{3}$

######  Proposition 1.

If $I^{j}$ is a $j$-interval that is $(n,\theta,j)$-good, then for any
$m\in\bar{T}^{j}_{n}$ the following holds

  1. 1.

$\underset{x\in
I^{j}}{\inf}\mid\frac{\partial\varphi_{m}}{\partial{x}}\mid\geq\frac{\theta
mq_{n}^{(j)}}{e^{q_{n}^{(j)}}},$

  2. 2.

$\underset{x\in
I^{j}}{\sup}\mid\frac{\partial^{2}\varphi_{m}}{\partial{x^{2}}}\mid\leq Cm.$

###### Proof.

For any $m\in\mathbb{N}$, we have

$\displaystyle\varphi_{m}(x)=m+\text{Re}\bigr{(}\sum_{j=1}^{4}\sum_{n=1}^{\infty}\frac{X_{j}(m,n)}{e^{q_{n}^{(j)}}}e^{i2\pi
q_{n}^{(j)}x_{j}}\bigl{)},$

where

$X_{j}(m,n)=\frac{1-e^{i2\pi mq_{n}^{(j)}\alpha_{j}}}{1-e^{i2\pi
q_{n}^{(j)}\alpha_{j}}},$

The proof of Proposition 1 is straightforward from the following lemma and we
refer the reader to [1] for the details.

######  Lemma 4 ([1]).

We have the following inequalities:

  * •

For all $m\in\mathbb{N}$, $\mid X_{j}(m,n)\mid\leq m.$

  * •

For all $m<q_{n}^{(j)}$,$m\in\mathbb{N}$,$\mid X_{j}(m,n)\mid\leq
q_{n}^{(j)}.$

  * •

For any $m\leq\frac{q_{n+1}^{(j)}}{2}$, $\mid
X_{j}(m,n)\mid\geq\frac{2m}{\pi}.$

###### Proof of Lemma 4..

The proof of the first inequality is obvious. The others follow from the fact
that

$\mid X_{j}(m,n)\mid=\left|\frac{\sin(\pi mq_{n}^{(j)}\alpha_{j})}{\sin(\pi
q_{n}^{(j)}\alpha_{j})}\right|,$

and the fact that by the definition of denominators of best approximations,
$q_{n}^{(j)}$ satisfies

$\lVert q_{n-1}^{(j)}\alpha_{j}\rVert<\lVert l\alpha_{j}\rVert,\quad\mbox{
$\forall l<q_{n}^{(j)},l\neq q_{n-1}^{(j)}$},$

as well as

$\frac{1}{q_{n}^{(j)}+q_{n+1}^{(j)}}\leq\lVert
q_{n}^{(j)}\alpha_{j}\rVert<\frac{1}{q_{n+1}^{(j)}}.$

∎

∎

As an immediate corollary of Proposition 1 and the definitions (2 .4) and (2
.5) and Lemma 3, we get that

######  Corollary 2.

If $I^{j}$ is a $j$-interval that is $(n,\theta,j)$-good, then for any $t\in
T^{j}_{n}$ the following holds

  1. 1.

$r_{I^{j}}^{t}\geq\theta t^{1-\epsilon/100}$

  2. 2.

$S_{I^{j}}^{t}\geq\theta^{2}t^{1-\epsilon/50}$

##  5 From uniform stretch to absolutely continuous spectrum. Proof of Theorem
3

###  5.1 Decay of correlations on good intervals due to uniform stretch

In all this section, we consider a pair of function $(f,g)$ such that : $f$ is
a smooth coboundary with transfer function $\psi$; $g\in{C^{1}}(M)$ and is
supported inside $M_{\zeta}$. We define for every $j\in\\{1,\ldots,4\\}$

$\displaystyle\mathcal{N}_{0}(f,g)$
$\displaystyle:=\|\psi\|_{0}\|g\|_{0}\quad\text{and}$ (5 .1)
$\displaystyle\mathcal{N}_{1}(f,g,j)$
$\displaystyle:=(\|f\|_{0}+\|\psi\|_{0})\|g\|_{1,j}+(\|f\|_{1,j}+\|\psi\|_{1,j})\|g\|_{0}\,,$
(5 .2)

where $\|\cdot\|_{0}$ denotes the $C^{0}$ norm and
$\|\psi\|_{1,j}=\|\psi\|_{0}+\|\partial_{x_{j}}f\|_{0}$. We also define
$\mathcal{N}_{1}(f,g)=\max_{j\in\\{0,\ldots,4\\}}\mathcal{N}_{1}(f,g,j)$.

In [1] the uniform stretch estimates above good intervals as in Corollary 2
(with $\theta$ bounded from below) are used to prove equi-distribution of good
intervals and then mixing by a Fubini argument. For the proof of countable
Lebesgue spectrum, we look for square summable estimates on the decay of
correlations, and this is why we use observables that are coboundaries. We
refer to [2, Proposition 6.2] for the proof of the following proposition that
relates uniform stretching of the Birkhoff sums and the decay of correlations.

######  Proposition 2.

For any $j$-interval $I^{j}\in M$ with extremities $(a,s)$ and $(b,s)$ and for
any $t\in T_{n}^{j}$, we have

$\left\lvert\int_{I^{j}}f(T^{t}_{\alpha,\varphi}z)g(z)d\lambda(z)-\Delta(I^{j})\right\rvert\leq
C\left(\mathcal{N}_{0}(f,g)\frac{\lambda(I^{j})}{S^{t}_{I^{j}}}+\mathcal{N}_{1}(f,g,j)\frac{\lambda(I^{j})}{r^{t}_{I^{j}}}\right),$

where
$\Delta(I^{j})=\frac{g(a,s)\psi(T^{t}(a,s))}{\partial_{j}\varphi_{N(a,s)}(a)}-\frac{g(b,s)\psi(T^{t}(b,s))}{\partial_{j}\varphi_{N(b,s)}(b)}$.

######  Remark 1.

The only difference between Proposition 2 and Proposition 6.2 of [2], is the
use of the norm $\mathcal{N}_{1}(f,g,j)$ to capture the fact that the interval
of integration is in the direction of $x_{j}$. In [2] the special flow was
considered above a circular rotation and the intervals were as a consequence
all in the same direction. It is natural and transparent from the proof of
Proposition 6.2 of [2] that the derivative that appear in the estimate of the
integral $\int_{I^{j}}f(T^{t}_{\alpha,\varphi}z)g(z)d\lambda(z)$ is in the
$j$-direction.

As a direct consequence of Propositions 1 and 2 we have the following crucial
estimate on the decay of correlations of the flow $T^{t}_{\alpha,\varphi}$
(when, as assumed in all this section, one of the functions is a smooth
coboundary):

######  Corollary 3.

If $I^{j}$ is a $j$-interval that is $(n,t^{-1/4+\epsilon},j)$-good, and if
$t\in T_{n}^{j}$, it holds that

$\left\lvert\int_{I^{j}}f(T^{t}_{\alpha,\varphi}z)g(z)d\lambda(z)\right\rvert\leq
C\mathcal{N}_{1}(f,g,j)\lambda(I^{j})t^{-1/2-\epsilon}.$

###  5.2 Partial partitions into good intervals

From Lemma 2, every $t\in\mathbb{R}$ belongs to at least three intervals
$T_{n_{j}}^{j}$. Fix now $t$ such that $t\in T_{n}^{1}\bigcap T_{n}^{2}\bigcap
T_{n}^{3}$, the other cases being treatable in exactly a similar fashion.

######  Proposition 3.

Ê For any $\zeta>0$, if $t$ is sufficiently large, there exists a partial
partition of $M$, $\mathcal{G}_{t}$ such that the atoms of $\mathcal{G}_{t}$
are $j$-intervals that are $(n,t^{-1/4+\epsilon},j)$-good, with
$j\in\\{1,2,3\\}$ and $\mu(\mathcal{G}_{t}^{c}\cap M_{\zeta})\leq
t^{-1/2-\epsilon}$.

Note that we cannot take $j$ to be the same for all the intervals in
$\mathcal{G}_{t}$.

###### Proof.

We first consider the partial partition $\mathcal{P}_{1}$ of $\mathbb{T}$ into
intervals that are the connected components of
$\mathcal{W}(n,t^{-1/4+\epsilon},1)=\left\\{x\in\mathbb{T}^{1}/\\{q_{n}^{(j)}x\\}\in[\theta,\frac{1}{2}-\theta]\cup[\frac{1}{2}+\theta,1-\theta]\right\\}$.
Next we consider on $\mathbb{T}^{4}$ the partial partition $\mathcal{G}^{(1)}$
consisting of $1$-intervals of the form $I\times(x_{2},x_{3},x_{4})$ where $I$
ranges over all the intervals of $\mathcal{P}_{1}$ and where
$(x_{2},x_{3},x_{4})$ ranges over all points in $\mathbb{T}^{3}$. The union of
the atoms of $\mathcal{G}^{(1)}$ covers all $\mathbb{T}^{4}$ except for a set
$\mathcal{E}_{1}$ that is the union of the bands $x_{1}\in\Delta_{n,k}^{(1)}$
and $x_{1}\in\bar{\Delta}_{n,k}^{(1)}$, where
$\Delta_{n,k}^{(1)}=[\frac{k}{q_{n}^{(1)}}-\frac{t^{-1/4+\epsilon}}{q_{n}^{(1)}},\frac{k}{q_{n}^{(1)}}+\frac{t^{-1/4+\epsilon}}{q_{n}^{(1)}}]$
and $\bar{\Delta}_{n,k}^{(1)}=\frac{1}{2q_{n}^{(1)}}+\Delta_{n,k}$, and
$k\in\\{0,\ldots,q_{n}^{(1)}-1\\}$.

Next, we consider the partial partition $\mathcal{P}_{2}$ of $\mathbb{T}$ into
intervals that are the connected components of
$\mathcal{W}(n,t^{-1/4+\epsilon},2)$. Next, we take a partial partition
$\mathcal{G}^{(2)}$ of the set $\mathcal{E}_{1}$ into $2$-intervals that are
of the form $x_{1}\times I\times(x_{3},x_{4})$ where $I$ ranges over all
intervals of $\mathcal{P}_{2}$ and $(x_{3},x_{4})$ over all points in
$\mathbb{T}^{2}$ and $x_{1}\in\Delta$ where $\Delta$ ranges over all bands
$\Delta_{n,k}^{(1)}$ and $\bar{\Delta}_{n,k}^{(1)}$, for
$k\in\\{0,\ldots,q_{n}^{(1)}-1\\}$.

The union of the atoms of $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(2)}$ covers
all $\mathbb{T}^{4}$ except for a set $\mathcal{E}_{2}$ that is the union of
the bands $(x_{1},x_{2})\in\Delta\times\Delta^{\prime}$ where $\Delta$ ranges
over all the sets $\Delta_{n,k}^{(1)}$ and $\bar{\Delta}_{n,k}^{(1)}$, for
$k\in\\{0,\ldots,q_{n}^{(1)}-1\\}$ and $\Delta^{\prime}$ ranges over all the
sets ${\Delta}_{n,k}^{(2)}$ and $\bar{\Delta}_{n,k}^{(2)}$, for
$k\in\\{0,\ldots,q_{n}^{(2)}-1\\}$.

Finally, we consider the partial partition $\mathcal{P}_{3}$ of $\mathbb{T}$
into intervals that are the connected components of
$\mathcal{W}(n,t^{-1/4+\epsilon},3)$. Next, we take a partial partition
$\mathcal{G}^{(3)}$ of the set $\mathcal{E}_{2}$ into $3$-intervals that are
of the form $(x_{1},x_{2})\times I\times x_{4}$ where $I$ ranges over all
intervals of $\mathcal{P}_{3}$ and $x_{4}$ over all points in $\mathbb{T}$ and
$(x_{1},x_{2})$ over all points in the projection of $\mathcal{E}_{2}$ on the
first and second coordinates of $\mathbb{T}^{4}$.

We let $\mathcal{G}$ be the partial partition of $\mathbb{T}^{4}$ consisting
of the atoms of $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(2)}$ and
$\mathcal{G}^{(3)}$. The union of the atoms of $\mathcal{G}_{t}$ covers all
$\mathbb{T}^{4}$ except for a set $\mathcal{E}_{3}$ that is the union of the
one dimensional bands
$(x_{1},x_{2},x_{3})\in\Delta\times\Delta^{\prime}\times\Delta^{\prime\prime}$
where $\Delta$ ranges over all the sets $\Delta_{n,k}^{(1)}$ and
$\bar{\Delta}_{n,k}^{(1)}$, for $k\in\\{0,\ldots,q_{n}^{(1)}-1\\}$, and
$\Delta^{\prime}$ ranges over all the sets ${\Delta}_{n,k}^{(2)}$ and
$\bar{\Delta}_{n,k}^{(2)}$ for $k\in\\{0,\ldots,q_{n}^{(2)}-1\\}$, and
$\Delta^{\prime\prime}$ over all the sets ${\Delta}_{n,k}^{(3)}$ and
$\bar{\Delta}_{n,k}^{(3)}$ for $k\in\\{0,\ldots,q_{n}^{(3)}-1\\}$. Clearly the
set $\mathcal{E}_{3}$ has a Haar measure on $\mathbb{T}^{4}$ bounded by
$4^{3}t^{-3/4+3\epsilon}=o(t^{-1/2-\epsilon})$.

To conclude, we want to extend the partial partition $\mathcal{G}$ to the
phase space $M$ of the special flow $\\{T^{t}_{\alpha,\varphi}\\}$. It is here
that we need to use the sets $M_{\zeta}$. For a fixed $\zeta>0$, we define for
sufficiently large $t$, the partial partition $\mathcal{G}_{t}$ as follows.
For every interval $I\in\mathcal{G}$ we include in $\mathcal{G}_{t}$ all the
intervals of the form $I\times s$ that satisfy $I\times s\cap
M_{\zeta}\neq\emptyset$ and $I\times s\cap M_{\zeta/2}^{c}=\emptyset$. The
latter condition insures that all the intervals we end up including in
$\mathcal{G}_{t}$ are all disjoint since they were disjoint in $\mathcal{G}$.
The former condition insures that $\mu(\mathcal{G}_{t}^{c}\cap
M_{\zeta})\leq\lambda_{4}(\mathcal{G})=o(t^{-1/2-\epsilon})$. ∎

###  5.3 Square summable decay of correlations. Proof of Theorem 3

Recall that by Proposition 3 we have a partial partition $\mathcal{G}_{t}$ in
good intervals of various directions. Since $\mu(\mathcal{G}_{t}^{c}\cap
M_{\zeta})\leq t^{-1/2-\epsilon}$, and since $g$ vanishes on $M_{\zeta}^{c}$,
we can apply the corollary in the various directions depending on the interval
of the partial partition that we consider, we get by Fubini the required
decay. More precisely, for every atom of $\mathcal{G}_{t}$ that is a good
interval $I$ in some direction $j$, we have from Corollary 3 that

$\left\lvert\int_{I}f(T^{t}_{\alpha,\varphi}(z))g(z)d\lambda(z)\right\rvert\leq
C\mathcal{N}_{1}(f,g)\lambda(I)t^{-1/2-\epsilon}.$

By Fubini, we get

$\left\lvert\int_{\mathcal{G}_{t}}f(T^{t}_{\alpha,\varphi}(z))g(z)d\mu(z)\right\rvert\leq
C\mathcal{N}_{1}(f,g)t^{-1/2-\epsilon},$

which implies, since $g$ vanishes on $M_{\zeta}^{c}$ that

$\left\lvert\int_{\mathcal{G}_{t}\cup
M_{\zeta}^{c}}f(T^{t}_{\alpha,\varphi}(z))g(z)d\mu(z)\right\rvert\leq
C\mathcal{N}_{1}(f,g)t^{-1/2-\epsilon},$

which implies, since $\mu(\mathcal{G}_{t}^{c}\cap M_{\zeta})\leq
t^{-1/2-\epsilon}$ that

$\left\lvert\int_{M}f(T^{t}_{\alpha,\varphi}(z))g(z)d\mu(z)\right\rvert\leq
C\mathcal{N}_{1}(f,g)t^{-1/2-\epsilon},$

as required. $\Box$

##  6 Infinite Lebesgue Spectrum

###  6.1 Criterion for Infinite Lebesgue Spectrum

In this subsection, we state a version adapted to our context of the criterion
for countable Lebesgue spectrum that was proved in [2]. For a multi-interval
$J=I_{1}\times\ldots\times I_{4}\subset\mathbb{T}^{4}$, let $T_{J}$ be the
maximal real number such that $T^{t}(J,0)\cap(J,0)=\emptyset$ for every
$|t|<T_{J}$.

For $T\leq T_{J}$ we define the tower above $J$

$R_{J}^{T}:=\bigcup_{t\in(-T,T)}F^{t}(J,0),$

and define the flow-box $F_{J}^{T}$ above $J$ with range $R_{J}^{T}$ as

$F^{T}_{J}(x,t)=T^{t}(x,0)\,,\,\,\text{ \rm for all }(x,t)\in J\times(-T,T).$

The flow-box $F_{J}:=F^{T_{J}}_{J}$ will be called a maximal flow-box over the
basis $J\subset M$.

Given a flow-box $F_{J}^{T}$, we define, for any $\zeta>0$, the set
$S^{T}_{\zeta}(J)\subset\mathbb{R}$ as follows

$S^{T}_{\zeta}(J):=\\{t\in(-T,T)\;:\;T^{t}(J)\cap
M_{\zeta}^{c}=\emptyset\\}\,.$

By definition we have that $S^{T}_{\zeta}(J)$ is an open subset (which in
general may be empty).

We can now define the functions supported on flow-boxes that we will be
working with.

######  Definition 2.

Given a flow-box $F_{J}^{T}$ and constants $C,\zeta>0$, we define
$\mathcal{G}(J,T,C,\zeta)$ to be the class of all functions $g\in
C^{\infty}(M)$, that vanish outside $R^{T}_{J}$, while on $R^{T}_{J}$ they are
defined as

$g\left(F^{T}_{J}(x,t)\right):=\chi_{J}(x)\psi(t)\,,\quad\text{ \rm if
}(x,t)\in J\times(-T,T)\,,\\\ $

with $\psi\in C^{\infty}_{0}(\mathbb{R},\mathbb{R})$, $\psi$ vanishes outside
$S^{T}_{\zeta}(J)$, and has $C^{1}$ norm bounded above by $C$, and
$\chi_{J}\in C_{0}^{\infty}(J)$ such that
$\int_{J}\chi^{2}_{J}d\lambda_{J}=1$, and

$\|\chi_{J}\|_{0}\leq C\lambda_{4}(J)^{-1/2},\quad\|\chi_{J}\|_{1,j}\leq
C\lambda_{4}(J)^{-1/2}|I_{j}|^{-1}.$ (6 .1)

The class $\mathcal{F}(J,T,C,\zeta)$ is the subset of
$\mathcal{G}(J,T,C,\zeta)$ consisting of smooth coboundaries.

The general criterion for countable Lebesgue spectrum stated in [2] implies in
our context the following. We use the notation

$\langle f\circ T^{t},g\rangle=\int_{M}f\circ
T^{t}_{\alpha,\varphi}(z)g(z)d\mu(z)\,.$

######  Theorem 4.

Assume $\\{T^{t}_{\alpha,\varphi}\\}$ has an absolutely continuous maximal
spectral type. If there exists a sequence of multi-intervals $J_{n}$ such that
$\lim\lambda(J_{n})=0$ and if for any $T>0$, $C>0$ and $\zeta>0$, for any
family $\\{(f_{n},g_{n})\\}$ of pair of functions such that
$f_{n}\in\mathcal{F}(J_{n},T,C,\zeta)$ and
$g_{n}\in\mathcal{G}(J_{n},T,C,\zeta)$, we have

$Ê\inf_{n}\int_{\mathbb{R}\setminus[-T_{J_{n}},T_{J_{n}}]}|\langle f_{n}\circ
T^{t},g_{n}\rangle|^{2}dt=0\,.$ (6 .2)

Then the flow $\\{T^{t}_{\alpha,\varphi}\\}$ has countable Lebesgue spectrum.

The statement of Theorem 4 is almost identical to Theorem 6 of [2], with this
important difference that in the class of functions that we consider is more
general in that we replaced in (6 .1) the condition

$\|\chi_{J}\|_{1}\leq C\lambda_{4}(J)^{-1/2-1}$

that was used in [2] by a less stringent and more precise one that
distinguishes between $C^{1}$ norms according to the direction in which the
derivatives are considered. Our condition is the natural one to control the
derivatives of a function supported inside $J$ and with $L^{2}$ norm bounded
away from $0$ and $\infty$. It is important for us to have a differentiated
control of the derivatives of the observables along different directions
because when we will show that the decay condition (6 .2) holds for the flow
$\\{T^{t}_{\alpha,\varphi}\\}$, we will use mixing estimates for intervals in
various directions, and these estimates naturally involve derivatives along
the direction of the interval as stated in Proposition 2.

In [2], the criterion for CILS was stated for general flow-boxes above multi-
intervals, but it was used for flows for which the decay of correlations did
not distinguish between various directions, a more precise statement such as
the one given here was not necessary.

######  Remark 2.

In [2], $M_{\zeta}$ consisted of points in $M$ that avoid a neighborhood of
the ceiling function, but that also avoid the singularity of the flow. For the
latter reason if one wanted to have the range of the flow-box above an
interval $J$ to spend most of its time in $M_{\zeta}$, then it was necessary,
in addition to taking $\zeta$ small, to start with an interval $J$ whose first
iterates avoid the neighborhood of the singularity. In our case, there is no
singularity and we do not need any extra assumption on the multi-interval $J$
besides its measure going to $0$.

######  Remark 3.

Compared to [2], we dropped the unnecessary condition on higher derivatives
since only the $C^{1}$ norms of the observables play a role in the decay of
correlations estimates.

The proof of Theorem 4 will be given in Section 7. As in [2], the proof will
be based on an abstract criterion that is very much the same as in Theorem 5
in [2]. For completeness, we do include a proof of the abstract criterion,
that is slightly simpler than the one given in [2]. For the rest of the proof
of Theorem 4, we follow exactly the same steps as in the proof of Theorem 5 in
[2], and we only insist on the differences that are imposed by the
differentiated control of the derivatives of the observables along different
directions.

###  6.2 Verification of the criterion for $\\{T^{t}_{\alpha,\varphi}\\}$

We prove below that the hypotheses of Theorem 4 are verified for
$\\{T_{\alpha,\varphi}^{t}\\}$. First of all, we know from Theorem 3 that
$\\{T^{t}_{\alpha,\varphi}\\}$ has an absolutely continuous maximal spectral
type.

We consider the following family of maximal flow boxes. For
$j\in\\{0,\ldots,4\\}$, let $J_{j,n}=[1/(8q_{n}^{(j)}),1/(4q_{n}^{(j)})]$ and
take $J_{n}=J_{1,n}\times\ldots\times J_{4,n}$. Note that by Lemma 3, it holds
that $T_{J_{n}}\geq q_{n}^{(1)}q_{n}^{(2)}q_{n}^{(3)}q_{n}^{(4)}$.

######  Theorem 5.

For any $T>0$, $C>0$, and $\zeta>0$, for any sequence of pair of functions
$\\{(f_{n},g_{n})\\}$ such that $f_{n}\in\mathcal{F}(J_{n},T,C)$ and
$g_{n}\in\mathcal{G}(J_{n},T,C,\zeta)$ we have

$\lim_{n\to 0}\int_{\mathbb{R}\setminus[-T_{J_{n}},T_{J_{n}}]}|\langle
f_{n}\circ T_{\alpha,\varphi}^{t},g_{n}\rangle|^{2}dt=0\,.$

###### Proof.

Having fixed $T>0$, we have that $T\in(0,T_{J_{n}}^{1/2})$ for $n$
sufficiently large. Let $|t|\geq T_{J_{n}}$. WLOG we can assume $t>0$ since
the argument for $t<0$ is similar.

Observe that the function $g_{n}$ is supported in $F^{T}_{J_{n}}\cap
M_{\zeta}$. Next, observe that there exists a set $A_{n}$ that is a disjoint
union of sets of the form $(R^{k}_{\alpha}J_{n},s)\subset M,|k|\leq 10T$ (by
Lemma 3) such that

$F^{T}_{J_{n}}\cap M_{\zeta}\subset A_{n},\quad\mu(A_{n})\leq
2\mu(F^{T}_{J_{n}}).$

Since $g_{n}$ vanishes on $A_{n}^{c}$, it is enough to prove bounds on

$\int_{A_{n}}f_{n}\circ T^{t}_{\alpha,\varphi}(z)g_{n}(z)d\mu(z)\,.$

First, we consider the case of $t\in[T_{J}/2,q^{(1)}_{n+1}/(n+1)]\subset
T_{n}^{1}$. We want to use the estimate in Proposition 2.

######  Claim 1.

The set $A_{n}$ is a disjoint union of $1$-intervals that are
$(n,1/50,1)$-good.

###### Proof.

Follows directly from the fact that $A_{n}$ is a disjoint union of sets of the
form $(R^{k}_{\alpha}J_{n},s)\subset M,|k|\leq 10T$. ∎

Consider any $1$-interval $I$ in the decomposition of $A_{n}$ given by Claim
1. Observe that since $f_{n},g_{n}\in\mathcal{G}(J_{n},T,C)$ (see Definition
2) then

$\mathcal{N}_{0}(f_{n},g_{n})\leq
C|J_{n}|^{-1},\quad\mathcal{N}_{1}(f,g,1)\leq Cq_{n}^{(1)}|J_{n}|^{-1}\leq
t^{\epsilon/100}\lambda_{4}(J)^{-1}.$

Since $f_{n}\in\mathcal{F}(J_{n},T,C)$, Propositions 1 and 2 and the fact that
the $1$-interval $I$ is $(n,1/50,1)$-good, imply

$\displaystyle\left\lvert\int_{I}f(T^{t}_{\alpha,\varphi}z)g(z)d\lambda(z)\right\rvert$
$\displaystyle\leq
C\\{\mathcal{N}_{0}(f,g))\frac{\lambda(I)}{S^{t}_{I}}+\mathcal{N}_{1}(f,g,1)\frac{\lambda(I)}{r^{t}_{I}}\\},$
$\displaystyle\leq Ct^{-1+2\epsilon}|J_{n}|^{-1}\lambda(I).$

Integrating over $I$, we get by Fubini

$\displaystyle\left\lvert\int_{M}f(T^{t}_{\alpha,\varphi}z)g(z)d\mu(z)\right\rvert=\left\lvert\int_{A_{n}}f(T^{t}_{\alpha,\varphi}z)g(z)d\mu(z)\right\rvert$
$\displaystyle\leq Ct^{-1+2\epsilon}|J_{n}|^{-1}\mu(A_{n})$ $\displaystyle\leq
Ct^{-1+2\epsilon}.$ (6 .3)

For $t>q^{(1)}_{n+1}/(n+1)$, we use Corollary 3 and the fact that
$\mathcal{N}_{1}(f_{n},g_{n})\leq{(q^{(1)}_{n+1})}^{\epsilon/100}$ to see that

$\left\lvert\int_{M}f(T^{t}_{\alpha,\varphi}z)g(z)d\mu\right\rvert\leq
t^{-1/2-\epsilon/2}.$ (6 .4)

In conclusion, we have proved that for $|t|\geq T_{J_{n}}$

$\left\lvert\int_{M}f(T^{t}_{\alpha,\varphi}z)g(z)d\mu\right\rvert\leq
Ct^{-1/2-\epsilon/2}.$

Squaring and integrating and using $T_{J_{n}}\to\infty$ as $n\to\infty$ we get
the desired decay. ∎

###  6.3 Proof of Theorem 2

The proof follows directly from Theorems 3, 4, and 5. $\Box$

##  7 The decay of correlations criterion for Infinite Lebesgue Spectrum

###  7.1 The general abstract criterion

Our criterion for countable Lebesgue spectrum of smooth flows is based on the
following abstract criterion that gives lower bounds on the multiplicity of a
strongly continuous one-parameter unitary group
$\\{\Phi_{t}\\}_{t\in\mathbb{R}}$ on on a separable Hilbert space $H$ with
absolutely continuous spectrum. The criterion is essentially the same as in
Theorem 5 in [2]. The proof that we give of this theorem is slightly simpler
than the one in [2].

Before we state the criterion, recall that the spectral theorem asserts that
for every $f\in H$, there exists a positive spectral measure $\nu_{f}$ on
$\mathbb{R}$ such that $\langle
f\circ\Phi_{t},f\rangle=\int_{\mathbb{R}}e(t\theta)d\nu_{f}(\theta)$. Since we
assume that $\\{\Phi_{t}\\}$ has absolutely continuous spectrum, we have that
$\nu_{f}$ is given by a positive density function with respect to Lebesgue
measure.

Given a measurable set $C\subset\mathbb{R}$ and an $L^{2}$ function $G$ on
$\mathbb{R}$, we denote by $\|G\|_{L^{2}(C)}$ the $L^{2}$ norm of the
restriction of $G$ on $C$. Unless specified, $L^{2}$ norms will be considered
with respect to the Lebesgue measure $\lambda$ on $\mathbb{R}$.

######  Theorem 6.

For a fixed $n\in\mathbb{N}$, let us assume that for every bounded set
$C\subset\mathbb{R}\setminus\\{0\\}$ of positive Lebesgue measure there exists
$\epsilon_{n,C}>0$ such that the following holds. For every
$\epsilon\in(0,\epsilon_{n,C})$ there exist vectors $f_{1},\dots,f_{n}\in H$
such that

$\displaystyle\|\langle f_{i}\circ\Phi_{t},f_{j}\rangle\|_{L^{2}}$
$\displaystyle\leq\delta_{ij}+\epsilon\,,\quad\text{ for all
}i,j\in{1,\dots,n}\,;$ (7 .1) $\displaystyle\|F_{i}^{2}-1\|_{L^{2}(C)}$
$\displaystyle\leq\epsilon,\quad\text{ for all }i\in{1,\dots,n}\,,$ (7 .2)

where $F_{i}^{2}(\cdot)$ is the density of the spectral measure associated to
$f_{i}$.

Then the spectral type of $\\{\Phi_{t}\\}_{t\in\mathbb{R}}$ is Lebesgue with
multiplicity at least $n$.

###### Proof.

Since the maximal spectral type is absolutely continuous with respect to
Lebesgue, we can assume that the sequence $(\mu_{k})$ is given by
$\mu_{k}=\phi_{k}(\theta)\frac{1}{1+\theta^{2}}d\theta$ where $\phi_{k}$ are
the characteristic functions of nested measurable sets $C_{k}$ on
$\mathbb{R}$.

Let $\bigoplus_{k\in\mathbb{N}}H_{k}$ denote the orthogonal decomposition of
$H$ into cyclic sub-spaces of $\Phi_{t}$ such that for all $i\in\mathbb{N}$,
we have $H_{k}\simeq L^{2}(\mathbb{R},\mu_{k})$.

Let us assume by contradiction that the spectrum is not Lebesgue with
multiplicity at least $n$. Then, there exists a compact set $\widehat{C}$ of
positive Lebesgue measure such that $\phi_{k}=0$ on $\widehat{C}$ for every
$k\geq n$.

Take $C=\widehat{C}$, and $f_{1},..,f_{n}\in H$ be as in Theorem 6 with
$\epsilon\ll 1$ to be specified later.

Now for any $U\in L^{\infty}(\mathbb{R},\lambda)$ such that $\lVert
U\rVert_{\infty}\leq 1$, take $h=h_{C,U}\in L^{2}(\mathbb{R})$ to be the
inverse Fourier transform of $\chi_{C}U$:

$\chi_{C}(\theta)U(\theta)=\int_{\mathbb{R}}h(s)e(\theta s)ds.$

We have that $\|h\|_{L^{2}}\leq\lambda(C)$.

For every $i\in\\{1,\ldots,n\\}$, let $f_{i}^{1},f_{i}^{2},\ldots$ denote the
successive orthogonal projections of $f_{i}$ on the spectral decomposition
$\bigoplus L^{2}(\mathbb{R},\mu_{k})$. By the definition of the spectral
isomorphism the projections of $f_{i}\circ\Phi_{s}$ are given by
$e(s\cdot)f_{i}^{1}(\cdot),e(s\cdot)f_{i}^{2}(\cdot),\ldots$

For every $i\in\\{1,\ldots,n\\}$, introduce the function $\nu_{i}\in H$:

$\nu_{i}=\int_{\mathbb{R}}h(s)f_{i}\circ\Phi_{s}ds.$

Since $\mu_{k}$ vanishes on $C$ for every $k\geq n$, we have that $\nu_{i}$
projects on
$\chi_{C}Uf_{i}^{1}\oplus\chi_{C}Uf_{i}^{2}\oplus\ldots\oplus\chi_{C}Uf_{i}^{n-1}$.

We apply assumption (7 .1) to $(f_{i},f_{j})$ for any $i\neq
j\in\\{1,\ldots,n\\}^{2}$. By the Cauchy Schwartz inequality, we obtain

$\displaystyle<\nu_{i},f_{j}>$
$\displaystyle=\int_{\mathbb{R}}<h(s)f_{i}\circ\Phi^{s},f_{j}>ds$
$\displaystyle=\int_{\mathbb{R}}h(s)<f_{i}\circ\Phi^{s},f_{j}>ds$
$\displaystyle\leq\|h\|_{L^{2}}\|<f_{i}\circ\Phi^{s},f_{j}>\|_{L^{2}}$
$\displaystyle\leq\lambda(C)\epsilon.$

We compute the same quantity using the spectral identification

$\begin{split}\mid<\nu_{i},f_{j}>\mid&=\mid\int_{\mathbb{R}}<h(s)f_{i}\circ\Phi^{s},f_{j}>ds\mid\\\
&=\mid\int_{\mathbb{R}}\sum_{k=1}^{\infty}\chi_{C}(\theta)U(\theta)f^{k}_{i}(\theta)f^{k}_{j}(\theta)\phi_{k}(\theta)\frac{1}{1+\theta^{2}}d\theta\mid\\\
&=\mid\int_{C}U(\theta)(\sum_{k=1}^{n-1}f^{k}_{i}(\theta)f^{k}_{j}(\theta))\phi_{k}(\theta)\frac{1}{1+\theta^{2}}d\theta\mid.\end{split}$

Define

$C_{i,j}=\\{\theta\in
C:\sum_{k=1}^{n-1}f^{k}_{i}(\theta)f^{k}_{j}(\theta)>\epsilon^{1/10}\\}.$

Taking $U=\chi_{C_{i,j}}$ we get for $\epsilon>0$ sufficiently small that
$\lambda(C_{i,j})\leq\lambda(C)\epsilon^{0.85}$. Doing the same for negative
values we get that if $\epsilon>0$ is sufficiently small

$\lambda\left(\left\\{\theta\in C:\exists i\neq j\in\\{1,\ldots,n\\}^{2},\
|\sum_{k=1}^{n-1}f^{k}_{i}(\theta)f^{k}_{j}(\theta)|>\epsilon^{1/10}\right\\}\right)<\epsilon^{0.8}.$
(7 .3)

We now observe that

$F_{i}^{2}(\theta)=\sum_{k=1}^{\infty}|f^{k}_{i}(\theta)|^{2}\phi_{k}(\theta)\frac{1}{1+\theta^{2}}.$

Hence, the second assumption (7 .2) of the theorem implies that

$\|\sum_{k=1}^{n-1}|f^{k}_{i}(\theta)|^{2}\phi_{k}(\theta)\frac{1}{1+\theta^{2}}-1\|_{L^{2}}\leq\epsilon.$

This implies that

$\lambda\left(\left\\{\theta\in C:\exists i\in\\{1,\ldots,n\\},\
\sum_{k=1}^{n-1}|f^{k}_{i}(\theta)|^{2}\notin[1/2,2]\right\\}\right)<\epsilon^{0.9}.$
(7 .4)

Finally, conditions (7 .3) and (7 .4) imply that there exists $\theta_{0}$
such that the vectors

$v_{i}=(f^{1}_{1}(\theta_{0}),..,f^{n-1}_{i}(\theta_{0})),\quad
i\in\\{1,\ldots,n\\},$

satisfy for all $(i,j)\in\\{1,\ldots,n\\}^{2}$, $j\neq i$

$|(v_{i}|v_{j})|<\epsilon^{0.1},\quad(v_{i}|v_{i})\in[\frac{1}{2},2],$

where $(\cdot|\cdot)$ denotes the Euclidean scalar product on
$\mathbb{C}^{n-1}$.

Since this is impossible, we conclude that the spectrum is Lebesgue with
multiplicity at least $n$.∎

###  7.2 The proof of Theorem 4

We want to use the assumption of Theorem 4 to check the validity of the
criterion of Theorem 6. In this section, the proofs will be exactly the same
as the ones given in Section 7 of [2], except for (7 .6) where the control on
the derivatives of the compactly supported functions that we use in the
construction are given in the various directions of the base $J$. Notice that
the control that we require in (7 .6) is exactly the one that allows to
guarantee the fast decay of correlations that we obtained in Theorem 5.

First of all, we recall the following corollary of Theorem 6 that was proved
in [2].

######  Corollary 4.

Let us assume that for every $n\in\mathbb{N}$, for any even functions
$\omega_{1},\dots,\omega_{n}\in\mathcal{S}(\mathbb{R})$ (the Schwartz space),
and for any any $\epsilon>0$, there exists $f_{1},\dots,f_{n}\in H$ such that,
for all $i,j\in\\{1,\dots,n\\}$, we have

$\|\langle
f_{i}\circ\phi_{t},f_{j}\rangle-\frac{d^{2}}{dt^{2}}\omega_{i}\ast\omega_{i}(t)\delta_{ij}\|_{L^{2}(\mathbb{R})}\leq\epsilon\,.$

Then the spectral type of the strongly continuous one-parameter unitary group
$\phi_{\mathbb{R}}$ is Lebesgue with countable multiplicity.

The fact that Theorem 6 implies Corollary 4 is clear since the assumption of
the corollary allows to produce $f_{1},\dots,f_{n}\in H$ that satisfy (7 .1)
and (7 .2).

We will derive Theorem 4 from Corollary 4. Since we only control the decay of
correlations for functions in the classes $\mathcal{F}(J,T,C,\zeta)$ and
$\mathcal{G}(J,T,C,\zeta)$, we need a simple approximation lemma to
approximate the target functions $\omega_{1},\dots,\omega_{n}$ by functions
supported inside sets of the type $S^{T}_{\zeta}(J)$, that we take as is from
[2].

######  Lemma 5 ([2, Lemma 7.4]).

Let $\\{F_{J}\\}$ be a family of maximal flow-boxes. For every $\epsilon>0$,
and even function $\omega\in\mathcal{S}(\mathbb{R})$, there exist
$\tau:=\tau(\epsilon,\omega)>0$ with the following property. For every
$T\geq\tau$, there exist constants $C:=C(\epsilon,\omega,T)>0$ and
$\zeta:=\zeta(\epsilon,\omega,T)>0$ such that, for any $F_{J}\in\Phi$ with
$T_{J}>T$, there exists an even function $\psi\in C^{\infty}_{0}(-T,T)$
satisfying

  1. $(a)$

$\frac{d\psi}{dt}\in C^{\infty}_{0}(S^{T}_{\zeta}(J)\cap(-S^{T}_{\zeta}(J)))$;

  2. $(b)$

the $C^{2}$ norm is bounded above by C;

  3. $(c)$

$\|\frac{d^{2}}{dt^{2}}(\psi\ast\psi)-\frac{d^{2}}{dt^{2}}(\omega\ast\omega)\|_{L^{2}(\mathbb{R})}<\epsilon\,.$

###### Proof of Theorem 4.

Let us fix $\epsilon>0$ and any given number $n\in\mathbb{N}\setminus\\{0\\}$
of even Schwartz functions
$\omega_{1},\dots,\omega_{n}\in\mathcal{S}(\mathbb{R})$. Let
$\Phi=\\{F_{J}\\}$ be a family of maximal flow-boxes.

By Lemma 5 there exists $\tau>0$ and, for all $T>\tau$, there exists $\zeta>0$
(small) such that, for every multi-interval $J=I_{1}\times\ldots\times
I_{4}\subset\mathbb{T}^{4}$, with $T_{J}>\tau$, there exist even functions
$\psi_{i}\in C^{\infty}_{0}((-T,T)),i=1,\ldots,n,$ with the property that
$\frac{d\psi_{i}}{dt}\in C^{\infty}_{0}(S^{T}_{\zeta}(J))$ and with $C^{2}$
norm uniformly bounded above by a constant
$C^{\prime}:=C^{\prime}(\epsilon,\omega_{1},\ldots,\omega_{n},T)>0$, such that

$\|\frac{d^{2}}{dt^{2}}(\psi_{i}\ast\psi_{i})-\frac{d^{2}}{dt^{2}}(\omega_{i}\ast\omega_{i})\|_{L^{2}(\mathbb{R})}<\epsilon/2\,.$
(7 .5)

Let now $\chi^{(1)}_{J},\dots,\chi^{(n)}_{J}\in C^{\infty}(J)$ be functions
such that

$\int_{J}\chi^{(i)}_{J}\chi^{(j)}_{J}d\lambda=\delta_{ij}\,,\quad\text{for all
}i,j\in\\{1,\dots,n\\}\,,$

and such that for $j\in\\{1,\ldots,4\\}$

$\|\chi^{(i)}_{J}\|_{0}\leq
C^{\prime\prime}\lambda_{4}(J)^{-1/2},\quad\|\chi^{(i)}_{J}\|_{1,j}\leq
C^{\prime\prime}\lambda_{4}(J)^{-1/2}{|I_{j}|}^{-1},$ (7 .6)

(this is possible provided that the constant $C^{\prime\prime}$ is taken to be
larger than some constant that only depends on $n$).

Let $C>\max\\{C^{\prime},C^{\prime\prime}\\}$. For every
$i\in\\{1,\dots,n\\}$, let $f^{(i)}_{J}\in\mathcal{F}(J,T,C,\zeta)$ be the
function defined on the range $R^{T}_{J}$ of the flow-box map $F^{T}_{J}$ as

$f^{(i)}_{J}\circ
F^{T}_{J}(x,t):=\chi^{(i)}_{J}(x)\frac{d}{dt}\psi_{i}(t)\,,\quad\text{ \rm if
}\,\,(x,t)\in J\times(-T,T)\,,$

and defined as $f^{(i)}_{J}=0$ on $M\setminus R^{T}_{J}$.

We then compute the correlations. Let $T_{J}/2>\max\\{T,\tau/2\\}$. For all
$t\in[-T_{J},T_{J}]$ we have (since the functions $\psi_{1},\dots,\psi_{n+1}$
are all even)

$\displaystyle\langle f^{(i)}_{J}\circ T^{t},f^{(j)}_{J}\rangle$
$\displaystyle=\int_{J}\int_{-T}^{T}\chi^{(i)}_{J}(x)\chi^{(j)}_{J}(x)\frac{d\psi_{i}}{dt}(\sigma+t)\frac{d\psi_{j}}{dt}(\sigma)d\sigma
dx$
$\displaystyle=(\frac{d\psi_{i}}{dt}\ast\frac{d\psi_{j}}{dt})(t)\,\delta_{ij}=\frac{d}{dt^{2}}(\psi_{i}\ast\psi_{j})(t)\delta_{ij}\,.$

By the assumption of Theorem 4 that we are proving, if $\lambda_{4}(J)$ is
small enough, for every $i,j\in\\{1,\ldots,n\\}$ we have:

$\|\langle f^{(i)}_{J}\circ
T^{t},f^{(j)}_{J}\rangle\|_{L^{2}(\mathbb{R}\setminus[-T_{J},T_{J}])}\leq\epsilon/2\,.$

Note that, since the functions $\psi_{i}$ are supported in $[-T,T]$ and
$T<T_{J}/2$, we also have

$\frac{d}{dt^{2}}(\psi_{i}\ast\psi_{j})(t)\delta_{ij}=0,\text{ for
}t\in\mathbb{R}\setminus[-T_{J},T_{J}].$ (7 .7)

By putting together formulas (7 .5)–(7 .7), it follows that if
$\lambda_{4}(J)$ is small enough (hence $T_{J}$ is large enough), the
functions $f^{(i)}_{J}$, with $i\in\\{1,\dots,n\\}$, satisfy the assumptions
of Corollary 4:

$\|\langle f^{(i)}_{J}\circ
T^{t},f^{(j)}_{J}\rangle-\frac{d^{2}}{dt^{2}}(\omega_{i}\ast\omega_{j})\delta_{ij}\|_{L^{2}(\mathbb{R})}\leq\epsilon\,.$

It follows then by Corollary 4 that, under the hypotheses of Theorem 4, the
flow $\\{T^{t}\\}$ has countable Lebesgue spectrum, hence the argument is
completed. ∎

### Acknowledgments.

The first and second author are very grateful to Giovanni Forni, Adam
Kanigowski and Jean-Paul Thouvenot for enlightening discussions on this
subject.

## References

  * [1] B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems 22(2002), 437–468.
  * [2] B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum of countable multiplicity for conservative flows on the torus, preprint, arXiv:1609.03757 .
  * [3] G. Forni and C. Ulcigrai, Time-Changes of Horocycle Flows, J. Mod. Dynam. 6 (2012), 251–273.
  * [4] A. B. Katok, Spectral properties of dynamical systems with an integral invariant on the torus. Dokl. Akad. Nauk SSSR 223 (1975), 789–792.
  * [5] A V. Kochergin, On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus, Dokl. Akad. Nauk SSSR 205 (1972), 949-952.
  * [6] A. V. Kochergin, Mixing in special flows over a shifting of segments and in smooth flows on surfaces, Mat. Sb. (N.S.) , 96 (138) (1975), 471–502.
  * [7] A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR 93 (1953), 763–766.
  * [8] A. N. Kolmogorov, Théorie générale des systèmes dynamiques et mécanique classique, (French) Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1, (1957), 315–333.
  * [9] J.-C. Yoccoz. Petits diviseurs en dimension 1.Astérisque, 1982, Appendix 1.


# The second gap on complete self-shrinkers

Qing-Ming Cheng, Guoxin Wei and Wataru Yano  Qing-Ming Cheng
Department of Applied Mathematics, Faculty of Sciences, Fukuoka University,
814-0180, Fukuoka, Japan<EMAIL_ADDRESS>Guoxin Wei
School of Mathematical Sciences, South China Normal University, 510631,
Guangzhou, China<EMAIL_ADDRESS>Wataru Yano
Department of Applied Mathematics, School of Sciences, Fukuoka University,
814-0180, Fukuoka, Japan<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study complete self-shrinkers in Euclidean space and prove
that an $n$-dimensional complete self-shrinker in Euclidean space
$\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$, $S^{n}(\sqrt{n})$,
or $S^{k}(\sqrt{k})\times\mathbb{R}^{n-k}$, $1\leq k\leq n-1$, if the squared
norm $S$ of the second fundamental form, $f_{3}$ are constant and $S$
satisfies $S<1.83379$. We should remark that the condition of polynomial
volume growth is not assumed.

††footnotetext:  2001  Mathematics Subject Classification: 53C44,
53C42.††footnotetext: Key words and phrases: the second fundamental form,
self-shrinkers and mean curvature flow.††footnotetext: The first author was
partially supported by JSPS Grant-in-Aid for Scientific Research (B):
No.16H03937. The second author was partly supported by grant No. 11771154 of
NSFC, Guangdong Province Universities and Colleges Pearl River Scholar Funded
Scheme (2018), Guangdong Natural Science Foundation Grant No.2019A1515011451.

##  1\. Introduction

Let $X:M\rightarrow\mathbb{R}^{n+1}$ be a smooth $n$-dimensional immersed
hypersurface in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$.
One calls an immersed hypersurface $X:M\rightarrow\mathbb{R}^{n+1}$ a self-
shrinker if it satisfies:

$\mathbf{H}=-X^{\bot},$

where $\mathbf{H}$ denotes the mean curvature vector of $M$, $X^{\bot}$
denotes the orthogonal projection of $X$ onto the normal bundle of $M$.

It is well known that Huisken [12] [13] and Colding and Minicozzi [8] have
proved that if $M$ is an $n$-dimensional complete embedded self-shrinker in
$\mathbb{R}^{n+1}$ with $H\geq 0$ and with polynomial volume growth, then $M$
is isometric to either the hyperplane $\mathbb{R}^{n}$, the round sphere
$S^{n}(\sqrt{n})$, or a cylinder $S^{m}(\sqrt{m})\times\mathbb{R}^{n-m}$,
$1\leq m\leq n-1$. For $n=1$, see Abresch and Langer [1].

######  Remark 1.1.

As one knows that self-shrinkers play an important role in the study of the
mean curvature flow because they describe all possible blow up at a given
singularity of a mean curvature flow.

On the other hand, Le and Sesum [14] and Cao and Li [2] have proved that if
$M$ is an $n$-dimensional complete self-shrinker with polynomial volume growth
and $S\leq 1$ in Euclidean space $\mathbb{R}^{n+1}$, then $M$ is isometric to
either the hyperplane $\mathbb{R}^{n}$, the round sphere $S^{n}(\sqrt{n})$, or
a cylinder $S^{m}(\sqrt{m})\times\mathbb{R}^{n-m}$, $1\leq m\leq n-1$. Ding
and Xin [9] have studied the second gap on the squared norm of the second
fundamental form and they have proved that if $M$ is an $n$-dimensional
complete self-shrinker with polynomial volume growth in Euclidean space
$\mathbb{R}^{n+1}$, there exists a positive number $\delta=0.022$ such that if
$1\leq S\leq 1+0.022$, then $S=1$. Furthermore, Cheng and Wei [6] have proved

######  Theorem 1.1.

Let $X:M\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional complete self-
shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$. If the squared
norm $S$ of the second fundamental form is constant and satisfies

$S\leq 1+\frac{3}{7},$

then $X:M\rightarrow\mathbb{R}^{n+1}$ is isometric to one of the following:

  1. (1)

$\mathbb{R}^{n}$,

  2. (2)

a cylinder $S^{k}(\sqrt{k})\times\mathbb{R}^{n-k}$,

  3. (3)

the round sphere $S^{n}(\sqrt{n})$.

In [4], Cheng and Ogata have obtained the following results (cf. Ding and Xin
[9]).

######  Theorem 1.2.

Let $X:M\rightarrow\mathbb{R}^{3}$ be a $2$-dimensional complete self-shrinker
in $\mathbb{R}^{3}$. If the squared norm $S$ of the second fundamental form is
constant, then $X:M\rightarrow\mathbb{R}^{3}$ is isometric to one of the
following:

  1. (1)

$\mathbb{R}^{2}$,

  2. (2)

a cylinder $S^{1}(1)\times\mathbb{R}$,

  3. (3)

the round sphere $S^{2}(\sqrt{2})$.

The following conjecture is known:

Conjecture. Let $X:M\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional
complete self-shrinker in $\mathbb{R}^{n+1}$. If the squared norm $S$ of the
second fundamental form is constant, then $X:M\rightarrow\mathbb{R}^{n+1}$ is
isometric to one of the following:

  1. (1)

$\mathbb{R}^{n}$,

  2. (2)

a cylinder $S^{k}(\sqrt{k})\times\mathbb{R}^{n-k}$,

  3. (3)

the round sphere $S^{n}(\sqrt{n})$.

######  Remark 1.2.

According to the result of Cheng and Ogata [4], this conjecture has been
solved for $n=2$. Recently, Cheng, Li and Wei [3] have solved this conjecture
for $n=3$ under the condition that $f_{4}$ is constant by making use of the
generalized maximum principle due to Cheng and Peng [5].

For general $n$, since this problem is too difficult, one can consider the
special case:

Problem. Let $X:M\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional complete
self-shrinker in $\mathbb{R}^{n+1}$. If the squared norm $S$ of the second
fundamental form is constant, then $X:M\rightarrow\mathbb{R}^{n+1}$ is
isometric to one of the following:

  1. (1)

$\mathbb{R}^{n}$,

  2. (2)

a cylinder $S^{k}(\sqrt{k})\times\mathbb{R}^{n-k}$,

  3. (3)

the round sphere $S^{n}(\sqrt{n})$,

  4. (4)

$S\geq 2$.

In this paper, we prove the following:

######  Theorem 1.3.

Let $X:M\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional complete self-
shrinker in $\mathbb{R}^{n+1}$. If the squared norm $S$ of the second
fundamental form and $f_{3}$ are constants and $S$ satisfies

$S\leq 1.83379,$

then $M$ is isometric to one of the following:

  1. (1)

$\mathbb{R}^{n}$,

  2. (2)

a cylinder $S^{k}(\sqrt{k})\times\mathbb{R}^{n-k}$,

  3. (3)

the round sphere $S^{n}(\sqrt{n})$,

where $f_{3}=\sum_{j=1}^{n}\lambda_{j}^{3}$ and $\lambda_{j}$’s are principal
curvatures of $X:M\rightarrow\mathbb{R}^{n+1}$.

######  Remark 1.3.

In our theorem 1.3, we do not assume that complete self-shrinkers
$X:M\rightarrow\mathbb{R}^{n+1}$ have polynomial volume growth and it is known
that there are many complete self-shrinkers without polynomial volume growth.

##  2\. Preliminaries

In this section, we give some notations and formulas. Let
$X:M\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional self-shrinker in
$\mathbb{R}^{n+1}$. Let $\\{e_{1},\cdots,e_{n},e_{n+1}\\}$ be a local
orthonormal basis along $M$ with dual coframe
$\\{\omega_{1},\cdots,\omega_{n},\omega_{n+1}\\}$, such that
$\\{e_{1},\cdots,e_{n}\\}$ is a local orthonormal basis of $M$ and $e_{n+1}$
is normal to $M$. Then we have

$\omega_{n+1}=0,\ \ \omega_{in+1}=\sum_{j=1}^{n}h_{ij}\omega_{j},\ \
h_{ij}=h_{ji},$

where $h_{ij}$ denotes the component of the second fundamental form of $M$.
$\mathbf{H}=\sum_{j=1}^{n}h_{jj}e_{n+1}$ is the mean curvature vector field,
$H=|\mathbf{H}|=\sum_{j=1}^{n}h_{jj}$ is the mean curvature and
$II=\sum_{i,j}h_{ij}\omega_{i}\otimes\omega_{j}e_{n+1}$ is the second
fundamental form of $M$. The Gauss equations and Codazzi equations are given
by

(2.1) $R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk},$ (2.2) $h_{ijk}=h_{ikj},$

where $R_{ijkl}$ is the component of curvature tensor, the covariant
derivative of $h_{ij}$ is defined by

$\sum_{k=1}^{n}h_{ijk}\omega_{k}=dh_{ij}+\sum_{k=1}^{n}h_{kj}\omega_{ki}+\sum_{k=1}^{n}h_{ik}\omega_{kj}.$

Let

$F_{i}=\nabla_{i}F,\ F_{ij}=\nabla_{j}\nabla_{i}F,\ h_{ijk}=\nabla_{k}h_{ij},\
{\rm and}\ h_{ijkl}=\nabla_{l}\nabla_{k}h_{ij},$

where $\nabla_{j}$ is the covariant differentiation operator, we have

(2.3)
$h_{ijkl}-h_{ijlk}=\sum_{m=1}^{n}h_{im}R_{mjkl}+\sum_{m=1}^{n}h_{mj}R_{mikl}.$

The following elliptic operator $\mathcal{L}$ is introduced by Colding and
Minicozzi in [8]:

(2.4) $\mathcal{L}f=\Delta f-\langle X,\nabla f\rangle,$

where $\Delta$ and $\nabla$ denote the Laplacian and the gradient operator on
the self-shrinker, respectively and $\langle\cdot,\cdot\rangle$ denotes the
standard inner product of $\mathbb{R}^{n+1}$. By a direct calculation, we have

(2.5) $\mathcal{L}h_{ij}=(1-S)h_{ij},\ \ \mathcal{L}H=H(1-S),\ \
\mathcal{L}X_{i}=-X_{i},\ \ \mathcal{L}|X|^{2}=2(n-|X|^{2}),$ (2.6)
$\frac{1}{2}\mathcal{L}S=\sum_{i,j,k}h_{ijk}^{2}+S(1-S).$

If $S$ is constant, then we obtain from (2.6)

(2.7) $\sum_{i,j,k}h_{ijk}^{2}=S(S-1),$

hence one has either

(2.8) $S=0,\ \ {\rm or}\ \ S=1,\ \ {\rm or}\ \ S>1.$

We can choose a local field of orthonormal frames on $M^{n}$ such that, at the
point that we consider,

$h_{ij}=\left\\{\begin{aligned} \lambda_{i},\ \ \quad&\text{if}\quad i=j,\\\
0,\ \ \quad&\text{if}\quad i\neq j,\end{aligned}\right.$

then

$S=\sum_{i,j}h_{ij}^{2}=\sum_{i}\lambda_{i}^{2},$

where $\lambda_{i}$ is called the principal curvature of $M$. From (2.1) and
(2.3), we get

(2.9) $h_{ijij}-h_{jiji}=(\lambda_{i}-\lambda_{j})\lambda_{i}\lambda_{j}.$

By a direct calculation, we obtain

(2.10) $\sum_{i,j,k,l}h_{ijkl}^{2}=S(S-1)(S-2)+3(A-2B),$

where $A=\sum_{i,j,k}\lambda_{i}^{2}h_{ijk}^{2},\ \
B=\sum_{i,j,k}\lambda_{i}\lambda_{j}h_{ijk}^{2}$.

We define two functions $f_{3}$ and $f_{4}$ as follows:

$f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}=\sum_{j=1}^{n}\lambda_{j}^{3},\ \ \ \
f_{4}=\sum_{i,j,k,l}h_{ij}h_{jk}h_{kl}h_{li}=\sum_{j=1}^{n}\lambda_{j}^{4},$

Then, the following formulas can be found in [6]:

######  Lemma 2.1.

Let $X:M\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional self-shrinker in
$\mathbb{R}^{n+1}$. Then

(2.11) $\dfrac{1}{3}\mathcal{L}f_{3}=(1-S)f_{3}+2C,$ (2.12)
$\dfrac{1}{4}\mathcal{L}f_{4}=(1-S)f_{4}+(2A+B).$ (2.13)
$A-B\leq\frac{1}{3}(\lambda_{1}-\lambda_{2})^{2}tS^{2},$ (2.14)
$C^{2}\leq\dfrac{1}{3}(A+2B)tS^{2},$

where $C=\sum_{i,j,k}\lambda_{i}h_{ijk}^{2}$.

##  3\. Estimates for geometric invariants

In this section, we will give some estimates which are needed to prove our
theorem. From now on, we denote

$S-1=tS,$

where $t$ is a positive constant and $t<\frac{1}{2}$ if we assume that $S$ is
constant and $S>1$, then

$(1-t)S=1,\ \ \ \ \sum_{i,j,k}h_{ijk}^{2}=tS^{2}.$

Defining

$u_{ijkl}:=\frac{1}{4}(h_{ijkl}+h_{jkli}+h_{klij}+h_{lijk}),$

we have

$\sum_{i,j,k,l}h_{ijkl}^{2}\geq\sum_{i,j,k,l}u_{ijkl}^{2}+\frac{3}{2}(Sf_{4}-f_{3}^{2})$

according to Gauss equations (2.1).
Let

(3.1) $Sf\equiv
Sf_{4}-\bigl{(}f_{3}\bigl{)}^{2}=S\sum_{i}\lambda_{i}^{4}-\left(\sum_{i}\lambda_{i}^{3}\right)^{2}=\dfrac{1}{2}\sum(\lambda_{i}-\lambda_{j})^{2}\lambda_{i}^{2}\lambda_{j}^{2}.$

Since $S$ and $f_{3}$ are constant, we have

$\sum_{i}\lambda_{i}h_{iik}=0,\ \ \sum_{i}\lambda_{i}^{2}h_{iik}=0,\ \text{\rm
for any }\ k$

and

$\sum_{i}\lambda_{i}h_{iikl}=-\sum_{i,j}h_{ijk}h_{ijl},\ \
\sum_{i}\lambda_{i}^{2}h_{iikl}=-2\sum_{i,j}\lambda_{i}h_{ijk}h_{ijl},\
\text{\rm for any }\ k,l.$

Hence, one has

$\sum_{i,j,}h_{iijj}\lambda_{i}\lambda_{j}=-C,\ \
\sum_{i,j}h_{iijj}\lambda_{i}^{2}\lambda_{j}=-2B.$

Defining

$a_{ij}=\sum_{m}h_{im}h_{mj}-yh_{ij},$

with $y=\dfrac{f_{3}}{S}$, we have

$\sum_{i,j,k,l}u_{ijkl}h_{ij}h_{kl}=\sum_{i,j}\dfrac{1}{2}(h_{iijj}+h_{jjii})\lambda_{i}\lambda_{j}=-C.$
$\sum_{i,j,k,l}u_{ijkl}a_{ij}h_{kl}=\sum_{i,j}\dfrac{1}{2}(h_{iijj}+h_{jjii})(\lambda_{i}^{2}-y\lambda_{i})\lambda_{j}=-B-\dfrac{1}{2}A+yC.$

Hence, because of

$\sum_{i,j,k,l}\biggl{\\{}u_{ijkl}+\alpha(a_{ij}h_{kl}+h_{ij}a_{kl})+\beta
h_{ij}h_{kl}\biggl{\\}}^{2}\geq 0,$

we obtain

$\displaystyle\sum_{i,j,k,l}u_{ijkl}^{2}$
$\displaystyle\geq-2\alpha\sum_{i,j,k,l}u_{ijkl}(a_{ij}h_{kl}+h_{ij}a_{kl})-\alpha^{2}\sum_{i,j,k,l}(a_{ij}h_{kl}+h_{ij}a_{kl})^{2}$
$\displaystyle-2\beta\sum_{i,j,k,l}u_{ijkl}h_{ij}h_{kl}-\beta^{2}\sum_{i,j,k,l}(h_{ij}h_{kl})^{2}-2\alpha\beta\sum_{i,j,k,l}(a_{ij}h_{kl}+h_{ij}a_{kl})h_{ij}h_{kl}$
$\displaystyle=2\alpha(2B+A-2yC)-2\alpha^{2}Sf+2\beta C-\beta^{2}S^{2}$
$\displaystyle\geq 2\alpha(2B+A-2yC)-2\alpha^{2}Sf+\dfrac{C^{2}}{S^{2}}$

by taking $\beta=\dfrac{C}{S^{2}}$. Since $f_{3}$ is constant, we have from
Lemma 2.1

$tSf_{3}=2C$

and

(3.2) $\displaystyle Sf$
$\displaystyle=Sf_{4}-f_{3}^{2}=\dfrac{1}{2}\sum_{i,j}(\lambda_{i}-\lambda_{j})^{2}\lambda_{i}^{2}\lambda_{j}^{2}$
$\displaystyle=\dfrac{1}{2}\sum_{i,j}(h_{iijj}-h_{jjii})(\lambda_{i}-\lambda_{j})\lambda_{i}\lambda_{j}$
$\displaystyle=A-2B,$

that is,

(3.3) $Sf=Sf_{4}-f_{3}^{2}=A-2B.$

From

(3.4) $\sum_{i,j,k,l}h_{ijkl}^{2}=S(S-1)(S-2)+3(A-2B)$

and

$\sum_{i,j,k,l}h_{ijkl}^{2}=\sum_{i,j,k,l}u_{ijkl}^{2}+\frac{3}{2}(Sf_{4}-f_{3}^{2}),$

we have

(3.5) $S(S-1)(S-2)=\sum_{i,j,k,l}u_{ijkl}^{2}-\dfrac{3}{2}(A-2B).$

Since $S$ is constant, by making use of the generalized maximum principle due
to Cheng and Peng [5], we have that there exist a sequence $\\{p_{k}\\}$ in
$M$ such that

(3.6) $tSf_{4}\geq 2A+B,$

it follows from (3.2) that

$tf_{3}^{2}\geq(2-t)A+(1+2t)B.$

Thus, for $z\geq 0$, from Lemma 2.1, one has

$C^{2}\leq\dfrac{1}{3}(A+2B)tS^{2}.$ (3.7)
$\displaystyle\sum_{i,j,k,l}u_{ijkl}^{2}-\dfrac{3}{2}(A-2B)$
$\displaystyle\geq
2\alpha(2B+A-2yC)-2\alpha^{2}Sf+\dfrac{C^{2}}{S^{2}}-\dfrac{3}{2}(A-2B)$
$\displaystyle=2\alpha(2B+A)-(2\alpha^{2}+\dfrac{3}{2})(A-2B)+(-2\alpha
t+(1+z)\dfrac{t^{2}}{4})f^{2}_{3}-z\dfrac{C^{2}}{S^{2}}$ $\displaystyle\geq
2\alpha(2B+A)-(2\alpha^{2}+\dfrac{3}{2})(A-2B)$
$\displaystyle+(-2\alpha+(1+z)\dfrac{t}{4})\bigl{\\{}(2-t)A+(1+2t)B\bigl{\\}}-z\dfrac{t}{3}(A+2B)$
$\displaystyle=\bigl{\\{}2\alpha-2\alpha^{2}-\dfrac{3}{2}+(-2\alpha+(1+z)\dfrac{t}{4})(2-t)-z\dfrac{t}{3}\bigl{\\}}A$
$\displaystyle+\bigl{\\{}4\alpha+4\alpha^{2}+3+(-2\alpha+(1+z)\dfrac{t}{4})(1+2t)-z\dfrac{2t}{3}\bigl{\\}}B,$

where $-2\alpha+(1+z)\frac{t}{4}\geq 0$. By taking

$tz=\dfrac{8\alpha^{2}-8t\alpha+6+t(t+3)}{1-t},$

we have

(3.8) $\displaystyle
4\alpha+4\alpha^{2}+3+(-2\alpha+(1+z)\dfrac{t}{4})(1+2t)-z\dfrac{2t}{3}$
$\displaystyle=-2\alpha+2\alpha^{2}+\dfrac{3}{2}-(-2\alpha+(1+z)\dfrac{t}{4})(2-t)+z\dfrac{t}{3}$
$\displaystyle=2(1-t)\alpha+2\alpha^{2}+\dfrac{3}{2}-\dfrac{t(2-t)}{4}+tz\dfrac{3t-2}{12}$
$\displaystyle=2(1-t)\alpha+2\alpha^{2}+\dfrac{3}{2}-\dfrac{t(2-t)}{4}+\dfrac{8\alpha^{2}-8t\alpha+6+t(t+3)}{1-t}\dfrac{3t-2}{12}$
$\displaystyle=\dfrac{4\alpha^{2}+4(3-4t)\alpha+3+2t(4t-3)}{6(1-t)}.$

We take $\alpha$ such that

(3.9) $4\alpha^{2}+4(3-4t)\alpha+3+2t(4t-3)=0$

if $t\leq\dfrac{9-\sqrt{33}}{8}$. Thus, we have from (3.5), (3.7) and (3.9)
that $t\geq\dfrac{1}{2}$. It is impossible. Hence, we have

$t>\dfrac{9-\sqrt{33}}{8}.$

In this case, taking $\alpha=-\dfrac{3-4t}{2}$, we obtain

$2\alpha^{2}+2(3-4t)\alpha+\dfrac{3}{2}+t(4t-3)=-\dfrac{(3-4t)^{2}}{2}+\dfrac{3}{2}+t(4t-3)=-4t^{2}+9t-3,$
(3.10) $\displaystyle tS^{2}(2t-1)S$
$\displaystyle\geq\sum_{i,j,k,l}u_{ijkl}^{2}-\dfrac{3}{2}(A-2B)$
$\displaystyle\geq\dfrac{4t^{2}-9t+3}{3(1-t)}(A-B).$

For any $i,j$, we have

$-\lambda_{i}\lambda_{j}\leq\dfrac{1}{4}(\lambda_{i}-\lambda_{j})^{2}.$

Hence, we get, for $\lambda_{i}\lambda_{j}\leq 0$ and
$\lambda_{i}\lambda_{k}\leq 0$,

(3.11) $(|\lambda_{i}\lambda_{j}|+|\lambda_{i}\lambda_{k}|)^{3}\leq
4(|\lambda_{i}\lambda_{j}|^{3}+|\lambda_{i}\lambda_{k}|^{3})\leq(\lambda_{i}-\lambda_{j})^{2}\lambda_{i}^{2}\lambda_{j}^{2}+(\lambda_{i}-\lambda_{k})^{2}\lambda_{i}^{2}\lambda_{k}^{2}.$

For three different $i,j,k$, we know that at least one of
$\lambda_{i}\lambda_{j}$, $\lambda_{i}\lambda_{k}$ and
$\lambda_{j}\lambda_{k}$ is non-negative. Without loss of generality, we
assume $\lambda_{j}\lambda_{k}\geq 0$ and $\lambda_{i}\lambda_{j}\leq 0$,
$\lambda_{i}\lambda_{k}\leq 0$, then we get from (3.11) and (3.1)

(3.12)
$-\lambda_{i}\lambda_{j}-\lambda_{i}\lambda_{k}-\lambda_{j}\lambda_{k}\leq|\lambda_{i}\lambda_{j}|+|\lambda_{i}\lambda_{k}|\leq(Sf)^{\frac{1}{3}},$

and

(3.13) $-2\lambda_{i}\lambda_{j}\leq
2(|\lambda_{i}\lambda_{j}|^{3})^{\frac{1}{3}}\leq
2(\dfrac{1}{4}(\lambda_{i}-\lambda_{j})^{2}\lambda_{i}^{2}\lambda_{j}^{2})^{\frac{1}{3}}\leq
2\bigl{(}\frac{1}{4}Sf\bigl{)}^{\frac{1}{3}}.$

Since

(3.14) $\displaystyle 3(A-B)$
$\displaystyle=\sum_{i,j,k}(\lambda_{i}^{2}+\lambda_{j}^{2}+\lambda_{k}^{2}-\lambda_{i}\lambda_{j}-\lambda_{i}\lambda_{k}-\lambda_{j}\lambda_{k})h_{ijk}^{2}$
$\displaystyle=\sum_{i\neq j\neq k\neq
i}(\lambda_{i}^{2}+\lambda_{j}^{2}+\lambda_{k}^{2}-\lambda_{i}\lambda_{j}-\lambda_{i}\lambda_{k}-\lambda_{j}\lambda_{k})h_{ijk}^{2}$
$\displaystyle\ \ \ +3\sum_{i\neq
j}(\lambda_{i}^{2}+\lambda_{j}^{2}-2\lambda_{i}\lambda_{j})h_{iij}^{2},$

we conclude from (3.12) and (3.13),

(3.15) $\displaystyle 3(A-B)$
$\displaystyle\leq\biggl{(}S+2\biggl{(}\frac{1}{4}Sf\biggl{)}^{\frac{1}{3}}\biggl{)}\sum_{i,j,k}h_{ijk}^{2}=\biggl{(}S+2\biggl{(}\dfrac{Sf}{4}\biggl{)}^{\frac{1}{3}}\biggl{)}tS^{2}.$

From Lemma 2.1 and (3.6), one gets

(3.16) $Sf\leq\frac{A-B}{3(1-t)}.$

Since $S-1=tS$, we infer from (3.15) and (3.16)

(3.17) $\displaystyle 3(A-B)$
$\displaystyle\leq\biggl{(}S+2\biggl{(}\dfrac{A-B}{12(1-t)}\biggl{)}^{\frac{1}{3}}\biggl{)}tS^{2}.$

If we assume that $3(A-B)\leq a_{k}tS^{3}$, then we obtain from (3.17) that

(3.18) $3(A-B)\leq a_{k+1}tS^{3},$

where

$a_{k+1}=\biggl{(}1+2\biggl{(}\dfrac{a_{k}}{36}\frac{t}{1-t}\biggl{)}^{\frac{1}{3}}\biggl{)}.$

Let $a_{1}=2$ and $t<0.454682$, we get from the above equations

(3.19) $a_{7}\leq 1.67738,$

then

(3.20) $3(A-B)\leq 1.67738tS^{3}.$

From (3.10) and (3.20), we have

(3.21) $(2t-1)\geq\dfrac{4t^{2}-9t+3}{3(1-t)}\times\frac{1.67738}{3}.$

Then, we get

$(t-0.454682)(t-1.24897)\leq 0,$

it follows that $t\geq 0.454682$. It is a contradiction. Hence, we get $t\geq
0.454682$ and $S\geq 1.83379$. We complete the proof of Theorem 1.3.

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