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##  Appendix A Data Release

The Dragonfly Data Access page can be found at
https://www.dragonflytelescope.org/data-access.html. The page contains a list
of publicly released fields grouped by their release date. Each field has its
own download page with links to the appropriate papers to cite if you make use
of the data in your work. Registration is required to facilitate communication
of any updated data products, but is open to anyone.

The images for all DNGS targets were previously released in November 2018.
Please note that fields are named for the primary target galaxy and some
fields contain multiple galaxies of interest. The images for NGC 4565 were
released in June 2020. The remaining 11 DEGS targets have now been released as
well.

In the FITS file header, NFRAMES indicates the number of single lens 600
second exposures that passed through all quality checks in the pipeline to be
included in the final stacked image. REFZP is the median of the photometric
zeropoints of individual frames contributing to the final stacked image. We
recommend using MEDIANZP as the image zeropoint instead, which has been
calibrated for the final stacked image.

##  Appendix B Mock galaxy tests

Although our target selection criteria included a proxy for Galactic cirrus
avoidance, cirrus is present to varying degrees in all of our fields. In
response, we have masked the worst of the cirrus by hand and handle
smooth/slowly-varying cirrus as a component of the background. We expect that
splitting each galaxy into four quadrants to better measure the local
background should also help mitigate cirrus contamination and variability. At
the faint surface brightness limits we have reached, one may ask how well the
true light distribution can be recovered in the presence of cirrus. We
therefore carried out mock galaxy tests in several of our fields spanning the
variety of cirrus conditions seen in this survey.

A mock galaxy image was created using Imfit (Erwin, 2015), using model
parameters similar to NGC 4565 but with an additional Sérsic component for the
stellar halo. This halo is more steeply declining than many of the observed
stellar halos in our sample and may be more or less vulnerable to cirrus
contamination or confusion. The mock galaxy was injected into several
locations in a few fields with a variety of cirrus conditions, as well as one
field where there appears to be little variation in the cirrus. These
positions were selected manually rather than randomly in order to avoid bright
stars, other galaxies, or extremely poor cirrus conditions. The chosen
positions have similar local cirrus morphology as the DEGS target galaxies:
usually either smooth, flat cirrus or a “hole” in the cirrus (an area with
flat cirrus surrounded by more bright and often more structured cirrus).
Although the cirrus morphology may be similar, the degree of contamination is
generally greater than at the position of a typical DEGS target (in terms of
cirrus brightness and/or the separation from structured cirrus).

Figure 14: Recovery of a mock galaxy’s surface brightness profile at several
different positions with varying cirrus conditions in four DEGS fields. The
true profile (extracted directly from the model image) is shown as a dashed
black line in each panel. “Mock E” is not shown; it was placed too close to
the field boundary and suffered from greatly decreased signal-to-noise.

Surface brightness profiles for each injected mock galaxy were extracted
following the procedure described in Section 3.1, including the masking of
bright or structured cirrus. The recovered profiles are shown in Figure 14. In
general, the outer profile where the stellar halo component dominates tends to
be somewhat oversubtracted in the presence of cirrus. The local background is
likely overestimated due to increasingly bright cirrus with increasing
separation from the mock galaxy. For the test field where there appears to be
little to no structured cirrus (NGC 4013, bottom left panel), the mock
galaxy’s surface brightness profile was well recovered in all three test
locations. This case may be more reflective of the cirrus conditions at the
center of DEGS fields than the oversubtracted profiles seen in other test
fields, as cirrus interference is typically lowest and smoothest at the
position of the target galaxy due to our cirrus avoidance criteria (low
$F_{100\mu m}$ within 5 arcmin of the galaxy).

We included one worst case scenario (Mock D, orange points in the upper left
panel) where the mock galaxy lies partly on a cirrus filament aligned with the
major axis. This leads to overestimation of light in the stellar halo region
due to heavy contamination on one side of the midplane and undersubtraction of
the background on the other side. This could be similar to the impact of
cirrus on NGC 4634 but is not representative of any of the other DEGS
galaxies.

The systematic uncertainty in the background is not always great enough for
the observed mock profiles to be consistent with the “true” mock profile.
$\Delta\mathrm{bkgd}_{sys}$ is currently defined as the standard deviation of
background values measured within the bins of the background annulus (Equation
C3). The full span of values within the background annulus
($\mathrm{bkgd}_{max}-\mathrm{bkgd}_{min}$) may be a fairer metric to capture
the difficulty in determining the most suitable background value. This is one
of the biggest sources of systematic uncertainty in our study, along with that
associated with the CMLR used to convert colour and surface brightness
profiles into stellar mass density profiles. If we have underestimated
$\Delta\mathrm{bkgd}_{sys}$, the uncertainty on our final measurements such as
$f_{halo}$ would also be underestimated.

These tests suggest that our profiles are unlikely to be artificially
brightened at large radii due to cirrus contamination. If anything,
oversubtraction is more likely to occur. However, there are a few important
caveats to consider. We only considered one mock galaxy rather than a suite of
galaxies with varying discs and halos. Some of our galaxies have larger
angular sizes than the mock galaxy, and our ability to handle variations in
the background due to cirrus may be worse at larger scales. We only injected
mock galaxies into four of twelve DEGS fields, partly due to the lack of
locations free from galaxies and bright stars.

##  Appendix C Error calculations

In this appendix, we summarize all of the calculations pertaining to the
sources of error mentioned throughout Sections 3.1-3.2. The corresponding
steps in the analysis are briefly described alongside the calculations.

We used this identity when converting between linear quantities and their
logarithms:

$\Delta\log x=\log e\frac{\Delta x}{x}.$ (C1)

###  C.1 Background measurement

The background was measured as the median of unmasked pixels within the radial
bin (within a given galaxy quadrant) that was identified to have the flattest
or sufficiently flat local slope. The random error of the background
measurement for each galaxy quadrant was defined by the standard error of the
median:

$\Delta\textrm{bkgd}=\frac{\sigma(\textrm{pixels})}{\sqrt{\frac{\pi}{2}N}}.$
(C2)

The standard deviation of the median background measurements from all radial
bins was adopted as the systematic uncertainty of the background measurement:

$\Delta\textrm{bkgd}_{\textrm{sys}}=\sigma(\textrm{bkgd}_{\textrm{annulus}}).$
(C3)

###  C.2 Surface brightness profiles

For each band $\Lambda$ (either $g$ or $r$) we extracted four quadrant
profiles $f_{\Lambda,i}$ with random measurement errors $\Delta
f_{\Lambda,i}$, internally calculated by photutils.isophote.ellipse. The
background was measured separately for each quadrant. The four flux profiles
were then averaged, weighted at each radius by number of unmasked pixels,
$n_{\Lambda,i}$). The error of combined flux profile is:

$\Delta
f_{\Lambda}=\frac{\sqrt{\sum\limits_{i=1}^{4}{n}_{\Lambda,i}^{2}(\Delta{f}_{\Lambda,i}^{2}+\Delta\textrm{bkgd}_{\Lambda,i}^{2}+\Delta\textrm{bkgd}_{\Lambda,i,\textrm{sys}}^{2})}}{\sum\limits_{i=1}^{4}{n}_{\Lambda,i}}.$
(C4)

The flux profiles were then corrected for scattered light by fitting 2D
parametric models to the disc, extracting profiles from model images with and
without PSF convolution, and then subtracting the difference between the PSF-
convolved and unconvolved profiles from the observed flux profiles. Following
Szomoru et al. (2010, 2012), we neglected error associated with the goodness
of fit of the 2D model and uncertainty in individual parameters because these
errors would be difficult to determine while also being small compared to
other sources of error (such as the systematic error of the background
measurement). We did consider the random measurement errors calculated by
photutils.isophote.ellipse while extracting profiles from the model images
because it was very straightforward to incorporate into our error analysis.
The error in $f_{\textrm{model,PSF},\Lambda}$ and
$f_{\textrm{model,noPSF},\Lambda}$ was combined in quadrature with the result
of Equation B4:

$\Delta f_{\Lambda,\textrm{corr}}=\sqrt{\Delta f_{\Lambda}^{2}+\Delta
f_{\textrm{model,PSF},\Lambda}^{2}+\Delta
f_{\textrm{model,noPSF},\Lambda}^{2}}.$ (C5)

Next, the corrected flux profiles were converted into surface brightness
profiles. We neglect uncertainty in the zeropoint. The error of the flux
profiles was converted to magnitudes:

$\Delta\mu_{\Lambda}=2.5\log e\frac{\Delta
f_{\Lambda,\textrm{corr}}}{f_{\Lambda,\textrm{corr}}}.$ (C6)

The error in the $g-r$ colour profile was obtained by combining the individual
errors of the $g-$ and $r-$band profiles ($\mu_{g}$ and $\mu_{r}$) in
quadrature:

$\Delta(g-r)=\sqrt{\Delta\mu_{g}^{2}+\Delta\mu_{r}^{2}}.$ (C7)

Where the noisy outer colour profile was replaced with a fixed value (the
average over the outermost five reliable points), we estimated the uncertainty
by combining the standard deviation of the outermost five reliable points with
the mean error of those points:

$\Delta(g-r)_{\textrm{fixed}}=\sqrt{\sigma(g-r)_{\textrm{outer}}^{2}+\textrm{avg}(\Delta(g-r)_{\textrm{outer}})^{2}}.$
(C8)

###  C.3 Surface stellar mass density profiles

The equation used to calculate $\Sigma_{*}$ from the colour and surface
brightness profiles is worth including here to provide context to our
subsequent error calculations:

$\displaystyle\log\Sigma_{*}$
$\displaystyle=\log\Big{(}\frac{\textrm{M}_{*}}{\textrm{L}}\times\textrm{L}\times\frac{1}{\Omega^{2}}\Big{)}$
(C9)
$\displaystyle=\log\Big{(}\frac{\textrm{M}_{*}}{\textrm{L}}\Big{)}_{g}+0.4(M_{g,\odot}-M_{g})+\log\Big{(}\frac{1}{\Omega^{2}}\Big{)}$
$\displaystyle=\log\Big{(}\frac{\textrm{M}_{*}}{\textrm{L}}\Big{)}_{g}-0.4(\mu_{g}-[M-m])+0.4(5.11\>\textrm{mag})+\log\Big{(}\frac{1}{\Omega^{2}}\Big{)}$
$\displaystyle=2.029(g-r)-0.984-0.4(\mu_{g}-[M-m])+2.044+\log\Big{(}\frac{1}{\Omega^{2}}\Big{)}.$

The solar absolute magnitude ($M_{g,\odot}$) was obtained from Willmer (2018).
In addition to the propagated uncertainty of $(g-r)$ and $\mu_{g}$, we
included the error of the CMLR itself, $\Delta\textrm{CMLR}=0.15$ and
$\Delta\textrm{CMLR}_{\textrm{sys}}=0.3$ (Roediger & Courteau, 2015). We
combined these sources of error in quadrature:

$\Delta\log\Sigma=\sqrt{\Big{(}0.4\cdot\Delta\mu_{g,\textrm{corr}}\Big{)}^{2}+\Big{(}2.029\cdot\Delta(g-r)\Big{)}^{2}+\Delta\textrm{CMLR}^{2}+\Delta\textrm{CMLR}_{\textrm{sys}}^{2}}.$
(C10)

###  C.4 Outer masses, disc masses, and stellar halo mass fractions

The $\Sigma_{*}$ profiles were integrated from radii corresponding to various
empirical definitions of the stellar halo-dominated region to the outer extent
of the profile to determine the stellar mass in these regions. The area
reported by photutils.isophote.ellipse for each isophote, $a_{r}$, was rounded
to nearest pixel so we adopted $\Delta a_{r}=1\>\textrm{pixel}\>\forall\>r$.
The error in integrating the $\Sigma_{*}$ profile from a radius of $r_{i}$ to
$r_{f}$ was calculated as:

$\Delta\log\textrm{M}_{*,\textrm{outer}}=\frac{\sqrt{\sum\limits_{r=r_{i}}^{r_{f}}(a_{r}\Sigma_{*,r})^{2}\Big{(}(\frac{\Delta
a_{r}}{a_{r}})^{2}+(\frac{1}{\log
e}\Delta\log\Sigma_{*,r})^{2}\Big{)}}}{\sum\limits_{r=r_{i}}^{r_{f}}a_{r}\Sigma_{*,r}}$
(C11)

A broken exponential disc was fitted to the bright inner regions of the
$\Sigma_{*}$ profile in order to estimate the stellar mass contributed by the
disc in these outer regions. The uncertainty in the disc mass contribution
(found by integrating $\Sigma_{*,\textrm{disc}}$ from $r_{i}$ to $r_{f}$) was
estimated by using emcee to generate a large number of independent samples
from the posterior probability density function of the disc model parameters.
Each sample (or set of possible disc model parameters) was used to calculate
the stellar mass of the disc in the outer region,
$\textrm{M}_{*,\textrm{disc}}$. The $16-84$ interquartile range of the
ensemble of $\textrm{M}_{*,\textrm{disc}}$ was used to estimate the $1\sigma$
upper and lower limits for the $\textrm{M}_{*,\textrm{disc}}$ calculated using
the maximum likelihood disc model parameters.

The stellar halo masses, $\textrm{M}_{*,\textrm{halo}}$, were obtained by
subtracting $\textrm{M}_{*,\textrm{disc}}$ from
$\textrm{M}_{*,\textrm{outer}}$. Upper and lower error bounds for
$\textrm{M}_{*,\textrm{halo}}$ and $f_{\textrm{halo}}$ were calculated
according to the equations below. The S4G total stellar masses,
$\textrm{M}_{*}$, were assumed to have a combined uncertainty of 0.2 dex
(Querejeta et al., 2015). We note that as a larger value of
$\textrm{M}_{*,\textrm{disc}}$ would result in a lower value of
$\textrm{M}_{*,\textrm{halo}}$, $\Delta\textrm{M}_{*,\textrm{disc,upper}}$ was
used to calculate $\Delta\textrm{M}_{*,\textrm{halo,lower}}$ while
$\Delta\textrm{M}_{*,\textrm{disc,lower}}$ was used to calculate
$\Delta\textrm{M}_{*,\textrm{halo,upper}}$.

$\Delta\log\textrm{M}_{*,\textrm{halo}}=\frac{\sqrt{\Big{(}\textrm{M}_{*,\textrm{outer}}\cdot\Delta\log\textrm{M}_{*,\textrm{outer}}\Big{)}^{2}+\Big{(}\textrm{M}_{*,\textrm{disc}}\cdot\Delta\log\textrm{M}_{*,\textrm{disc}}\Big{)}^{2}}}{\Big{(}\textrm{M}_{*,\textrm{outer}}-\textrm{M}_{*,\textrm{disc}}\Big{)}}$
(C12)

$\Delta\log
f_{\textrm{halo}}=\sqrt{\Delta\log\textrm{M}_{*,\textrm{halo}}^{2}+\Delta\log\textrm{M}_{*}^{2}}$
(C13)

##  Appendix D Additional figures

The following pages contain figures gathering all of the final profiles and
images for each DEGS galaxy. The galaxies are sorted by increasing NGC number.
All panels on the left side of a figure share a common horizontal scale. The
extent of the top right and middle right panels is shown with a white box in
the lower right panel. Top left: $g$\- and $r$-band surface brightness
profiles. Error bars on points depict random error. The shaded envelopes
depict upper and lower limits considering random error and systematic
uncertainty in the sky background measurement. Middle left: $g-r$ colour
profile. As with the surface brightness profiles above, error bars on points
depict random error and the shaded envelope depicts upper and lower limits
considering random error and systematic uncertainty. The grey vertical lines
indicate the region where the average outer colour is determined. Beyond the
second grey vertical line, the average outer colour is favoured over the raw
observed colours due to large error and/or gaps in either the $g$\- or
$r$-band profile. The grey points show this average outer colour. Bottom left:
Stellar mass surface density profile, in units of
$\mathrm{M}_{\odot}/\mathrm{kpc}^{2}$. As with the profiles above, error bars
on points depict random error and the shaded envelope depicts upper and lower
limits considering random error and systematic uncertainty. The vertical grey
line is included to mark the radius beyond which the average outer colour is
used over the raw observed colour. Top right: A central cutout of the $r$-band
source-subtracted image (as produced using the mrf package; see Section 3 for
further details), centered on the target galaxy. Masked areas are shown in
white. The $r-$band surface brightness profile shown in the top left panel is
extracted from this image along elliptical isophotes fitted using Photutils
(Bradley et al., 2019). The ellipse fitting procedure is described in Section
3.1. A subset of the best fit elliptical isophotes are shown in black. The
outermost ellipses are labelled with their approximate semi-major axis length.
Middle right: A false colour composite matching the footprint of the central
cutout in the top right panel. We generated the false colour composite using
the Lupton et al. (2004) method as implemented in
astropy.visualization.make_lupton_rgb(), with $Q=7$ and stretch$=10$. The
three image colour channels are as follows: $R=r$, $G=0.5(r+g)$, $B=1.2g$.
Bottom right: A false colour composite of the full Dragonfly field, centered
on the target galaxy. The white box shows the extent of the central cutouts
shown above. A scale bar indicating one degree is included.

Figure 15: A detailed summary figure for DEGS galaxy NGC 3044. Please refer to
the beginning of Appendix D for a description of this figure. Figure 16: Same
as Figure 15 for NGC 3432. Figure 17: Same as Figure 15 for NGC 3501. Figure
18: Same as Figure 15 for NGC 3628. Figure 19: Same as Figure 15 for NGC 4010.
Figure 20: Same as Figure 15 for NGC 4013. Figure 21: Same as Figure 15 for
NGC 4307. Figure 22: Same as Figure 15 for NGC 4330. Figure 23: Same as Figure
15 for NGC 4437. Figure 24: Same as Figure 15 for NGC 4565. Figure 25: Same as
Figure 15 for NGC 4634. Figure 26: Same as Figure 15 for NGC 5907.


# Liouville ergodicity of linear multi-particle hamiltonian system with one
marked particle velocity flips

Lykov A. A., Malyshev V. A Faculty of Mechanics and Mathematics, Lomonosov
Moscow State University. Vorobievy Gory, Main Building, 119991, Moscow Russia,
<EMAIL_ADDRESS>
###### Abstract

We consider multi-particle systems with linear deterministic hamiltonian
dynamics. Besides Liouville measure it has continuum of invariant tori and
thus continuum of invariant measures. But if one specified particle is
subjected to a simple linear deterministic transformation (velocity flip) in
random time moments, we prove convergence to Liouville measure for any initial
state. For the proof it appeared necessary to study non-linear transformations
on the energy surface.

###### Contents

  1. 1 Introduction
  2. 2 Model and Main Results
  3. 3 Deterministic part - proof of covering theorem
    1. 3.1 Key definitions, notation and some intuition
    2. 3.2 Proof of closure theorem
      1. 3.2.1 Contraction property
      2. 3.2.2 Closure theorem
    3. 3.3 Proof of local covering theorem
    4. 3.4 Proof of theorem 1
  4. 4 Stochastic part - proof of convergence theorem
    1. 4.1 Embedded process
    2. 4.2 Proof of theorem 6
    3. 4.3 Proof of theorem 2
  5. 5 Appendix
    1. 5.1 Proof of lemma 2
    2. 5.2 Case $N=1$
    3. 5.3 Mixing subspace
    4. 5.4 Proof of lemma 3

##  1 Introduction

Ergodicity problem for hamiltonian multi-particle systems produced many deep
results. First of all, many examples of non-ergodic systems appeared - linear,
non-linear with additional integrals and close to them (KAM theory). One could
expect then that for generic hamiltonians one also has non-ergodic behaviour.
However, as far as we know, this is still an open difficult problem with many
partial results, see [20, 19], and, after the century history of the
ergodicity hypothesis it is reasonable to look for simpler alternative
approaches to it.

Namely, one could assume that any physical system has always some contact with
external world. Such contact can be of quite various extent: 1) all particles
can have contact with external world and stochastic behaviour (for example,
with dynamics of Glauber type), 2) only particles on the boundary, etc. But
then it is quite natural to ask - what is the minimal contact which definitely
provides ergodic behavior. Possible reformulation of the ergodicity hypothesis
could be the following: for generic system even the minimalistic contact
produces ergodic behaviour.

Here we consider an example of such minimalistic contact which consists, first
of all, in that we allow some contact with external world for only one
(marked) particle. In our earlier papers [11, 9] this particle was subjected
to some random force, that garantied convergence to Gibbs equilibrium. Here we
assume even less randomness. Namely, the marked particle is subjected to a
simple deterministic transformation (velocity flip, very popular in other
problems [6, 7, 8, 3, 5, 4]) but in discrete random time moments

$0<t_{1}<\ldots<t_{m}<\ldots$ (1)

Liouville measure is evidently invariant w.r.t. such dynamics on the energy
surface. We prove then that ergodicity in the stronger form holds - for any
initial state we have convergence to Liouville measure on the energy surface.

It is very interesting that it works even for the systems having the worst
possible non-ergodic behaviour - linear systems. The only price we pay is that
ergodicity holds not for any linear system but for almost any - this is purely
algebraic phenomenon which was discussed in our earlier papers and cannot be
avoided. It seems reasonable that the same result holds for non-linear systems
as well (following the common belief that the latter have better mixing
properties than linear systems).

The paper is naturally subdivided in two parts. The first part uses no
probability at all but only elaborate non-linear analysis to prove that the
trajectory visits all invariant tori and even any point - we had to use
coordinates on the energy surface where velocity flips are strongly non-
linear. Second part, on the contrary, essentially uses non-trivial parts of
Markov processes theory on continuous state space.

Now we come to rigorous definitions.

##  2 Model and Main Results

##### Hamiltonian dynamics

We consider the linear space

$L=\mathbb{R}^{2N}=\\{\psi=\left(\begin{array}[]{c}q\\\ p\end{array}\right):\
q=(q_{1},...,q_{N})^{T},\ p=(p_{1},...,p_{N})^{T},\
q_{i},p_{i}\in\mathbb{R}\\},$

where $T$ denotes transposition (thus $\psi$ is a column-vector). It can be
presented as the direct sum $L=l_{N}^{(q)}\oplus l_{N}^{(p)}$ of two
orthogonal (coordinate and momenta) spaces of dimension $N$ with the standard
scalar product in $\mathbb{R}^{2N}$

$(\psi,\psi^{\prime})_{2}=(q,q^{\prime})_{2}+(p,p^{\prime})_{2}=\sum_{i=1}^{N}(q_{i}q_{i}^{\prime}+p_{i}p_{i}^{\prime})$

We consider quadratic hamiltonian

$H(\psi)=\sum_{k=1}^{N}\frac{p_{k}^{2}}{2}+U(q),U(q)=\frac{1}{2}(q,Vq)_{2}$
(2)

where the matrix $V>0$ acting in $\mathbb{R}^{N}$ is assumed to be real and
positive definite (thus the particles cannot escape to infinity). This defines
hamiltonian system of linear ODE with $k=1,\ldots,N$

$\dot{q}_{k}=p_{k},\dot{p}_{k}=-\sum_{l=1}^{N}V_{kl}q_{l},$ (3)

For any $h>0$ define the constant energy surface

$\mathcal{M}_{h}=\\{\psi\in L:H(\psi)=h\\}$

Then $\mathcal{M}_{h}$ is a smooth manifold (ellipsoid) in $L$ of codimension
$1$.

##### Allowed hamiltonians

Define the mixing subspace

$L_{-}=L_{-}(V)=\\{\left(\begin{array}[]{c}q\\\ p\end{array}\right)\in
L:q,p\in l_{V}\\}$

where $l_{V}=l_{V,1}$ is the subspace of $\mathbb{R}^{N}$, generated by the
vectors $V^{k}e_{1},\ k=0,1,2\ldots$, where $e_{1},...,e_{N}$ is the standard
basis in $\mathbb{R}^{N}$.

Let $\mathbf{V}$ be the set of all positive-definite $(N\times N)$-matrices,
and let $\mathbf{V}^{+}\subset\mathbf{V}$ be the subset of matrices for which

$L_{-}(V)=L$ (4)

The set of $V\in\mathbf{V}$ such that their eigenvalues, denoted by
$\omega_{1}^{2},\ldots,\omega_{N}^{2}$, are independent over the field of
rational numbers, is denoted by $\mathbf{V}_{ind}$.

###### Lemma 1

The set $\mathbf{V}^{+}$ is open and everywhere dense (assuming topology of
$R^{\frac{N(N+1)}{2}}$) in $\mathbf{V},$ and the set
$\mathbf{V}^{+}\cap\mathbf{V}_{ind}$ is dense both in $\mathbf{V}^{+}$and in
$\mathbf{V}.$

See more in section 5.3.

##### Piecewise deterministic process

Assume that at time moments (1) the following deterministic transformation
$I:L\to L$ occurs: all $q_{k},p_{k}$ are left unchanged, except for $p_{1}$,
the sign of which becomes inverted

$p_{1}(t_{m}-0)\to p_{1}(t_{m})=-p_{1}(t_{m}-0),m\geq 1$

For example, one can consider $L$ as the phase space for $N$ identical point
particles in $\mathbb{R}$, with mass $m=1$, and real numbers $q_{i},p_{i}$ are
their coordinates and velocities (momenta). Then this transformation can be
interpreted as the elastic collision of the particle $1$ with a wall.
Alternatively, taking $dN$ instead of $N$, one can imagine $N$ particles in
$R^{d}$ where only one velocity component of particle 1 is flipped.
Reflections w.r.t. any hyperplane in $R^{d}$ could be considered quite
similarly.

In-between these moments the system evolves via hamiltonian dynamics (3).

With $(2N\times 2N)$-matrix

$A=\left(\begin{array}[]{cc}0&E\\\ -V&0\end{array}\right)$

the system (3) can be rewritten as

$\dot{\psi}=A\psi.$ (5)

and the solution $\psi(t)$ of (5) with initial vector $\psi(0)$ will be

$\psi(t)=e^{tA}\psi(0).$

For given sequence (1) the dynamics of our process is defined for $t_{m}\leq
t<t_{m+1}$ as

$\psi(t)=e^{A(t-t_{m})}Ie^{A\tau_{m}}Ie^{A\tau_{m-1}}I...Ie^{A\tau_{2}}Ie^{A\tau_{1}}\psi(0)$

where $\tau_{1}=t_{1},\tau_{2}=t_{2}-t_{1},...,\tau_{m}=t_{m}-t_{m-1},...$.
For any $t\geqslant 0$ define linear maps $L\to L$

$J(t)\psi=Ie^{tA}\psi,\ \psi\in L$

It is clear that $\mathcal{M}_{h}$ is invariant w.r.t. $J(t)$ for any $h>0$
and $t>0$. For any $\psi\in L$ and any integer $m\geqslant 1$ define the set
of states

$\mathcal{J}_{m}(\psi)=\\{J(\tau_{m})\ldots J(\tau_{1})\psi:\
0\leq\tau_{1},...,\tau_{m}\\}\subset\mathcal{M}_{h}$

which the system can visit at the $m$-th flip.

######  Theorem 1 (covering theorem)

Assume that $V\in\mathbf{V}^{+}\cap\mathbf{V}_{ind}$, then there exists
$m\geqslant 1$ such that for any $\psi\in L$ we have

$\mathcal{J}_{m}(\psi)=\mathcal{M}_{h}$

We introduce randomness by the following assumption

A0): positive random variables

$\tau_{1}=t_{1},\tau_{2}=t_{2}-t_{1},...,\tau_{n}=t_{n}-t_{n-1},...$

are assumed to be independent, identically distributed with measure
$P_{\tau}=\rho(s)ds$ where the density $\rho$ (w.r.t. Lebesgue measure $ds$)
is positive everywhere on $R_{+}$, and moreover the first moment
$E\tau_{1}<\infty$.

If, for example, $\tau_{i}$ have exponential distribution with the density
$\lambda\exp(-\lambda\tau),\ \lambda>0$, then it defines Markov process
$\psi(t)$ with right continuous deterministic trajectories and random jumps.
Such processes are often called piecewise deterministic Markov processes, see
for example [14]. At the same time, this can be considered as an example from
random perturbation theory, see [12] where the problem of invariant measures
is studied.

Let $\pi$ be Liouville measure on the energy surface, defined by the surface
form $d\sigma$ divided by $|\nabla H|$. It is well known that $\pi$ is
invariant w.r.t. hamiltonian dynamics and also w.r.t. velocity flips.

######  Theorem 2 (convergence theorem)

Assume that $V\in\mathbf{V}^{+}\cap\mathbf{V}_{ind}$. Then under assumption
$A0)$ for any initial $\psi(0)$ and any bounded measurable real function $f$
on $\mathcal{M}_{h}$ we have a.s.

$M_{f}(T)=^{def}\frac{1}{T}\int_{0}^{T}f(\psi(t))dt\to\pi(f)=^{def}\int_{\mathcal{M}_{h}}fd\pi$

We call this property Liouville ergodicity.

##  3 Deterministic part - proof of covering theorem

##### Plan of the proofs

In all assertions below we always assume that
$V\in\mathbf{V}^{+}\cap\mathbf{V}_{ind}$. Theorem 2 will follow from theorem
1. Theorem 1 will follow from the following weaker results.

######  Theorem 3 (closure theorem)

There exists $m\geqslant 1$ such that for all $\psi\in L$ we have

$\overline{\mathcal{J}_{m}(\psi)}=\mathcal{M}_{h}$

The covering and closure theorems have simpler local analogue. And moreover it
shows that the dimension of $\mathcal{J}_{m}(\psi)$ grows as $m$ for
$m=1,2,...,2N-1$. For exact formulation we need some definitions. For any
$\tau_{1},...,\tau_{m}>0$ denote
$J(\tau_{1},\ldots,\tau_{m})=J(\tau_{m})\ldots J(\tau_{1})$. It can be
considered as the mapping from $L$ to $\mathcal{M}_{h}$ for fixed
$(\tau_{1},\ldots,\tau_{m})$, but also $J(\tau_{1},\ldots,\tau_{m})\psi$ can
be considered as the map

$J_{m}^{\psi}:\Omega_{m}\rightarrow\mathcal{M}_{h},\quad\bar{\tau}=(\tau_{1},\ldots,\tau_{m})\mapsto
J(\tau_{1},\ldots,\tau_{m})\psi.$

from the $m$-dimensional orthant $\Omega_{m}=\\{(\tau_{1},\ldots,\tau_{m}):\
\tau_{i}>0,\ \ i=1,\ldots,m\\}=\mathbb{R}_{+}^{m},$ to $\mathcal{M}_{h},$ for
fixed $\psi\in\mathcal{M}_{h}$.

######  Theorem 4 (local covering theorem)

For any point $\psi\in\mathcal{M}_{h}$ and any $k=1,\ldots,2N-1$ the dimension
of $J_{k}^{\psi}\Omega_{k}$ is equal to $k$. Moreover, there exists open
subset $U\subset\Omega_{k}$ such that the mapping
$J_{k}^{\psi}:U\hookrightarrow\mathcal{M}_{h}$ is a smooth embedding

###  3.1 Key definitions, notation and some intuition

The sequence (1) completely defines the trajectory, or the path. Proof of
convergence and covering theorems could be obtained, in some sense, by
“summation” over all possible paths. For this we have to use various
coordinate systems on the energy surface. To get some intuition it is useful
to see how it works for easier cases $N=1,2$.

##### Action-angle coordinates

Let $v_{1},\ldots,v_{N}$ be the orthonormal eigenvectors of $V$ in
$\mathbb{R}^{N}$, and let $\omega_{1}^{2},\ldots,\omega_{N}^{2}$ be the
corresponding eigenvalues of $V$. For any $k=1,\ldots,N$ define $2N$-vectors
in $L$:

$Q_{k}=(v_{k},0)^{T},\quad P_{k}=(0,v_{k})^{T}$ (6)

The coordinates of $\psi\in L$ in the basis $Q_{1},P_{1},\ldots,Q_{N},P_{N}$
we denote by $(\tilde{q_{k}},\tilde{p_{k}})^{T}$, that is

$\psi=\sum_{k=1}^{N}\tilde{q}_{k}Q_{k}+\sum_{k=1}^{N}\tilde{p}_{k}P_{k}$ (7)

They correspond to coordinates and momenta of the independent
<<quasiparticles>>, that is one-dimensional oscillators, having energies
$\frac{r_{k}^{2}}{2}$ where

$r_{k}^{2}(\psi)=\tilde{p}_{k}^{2}+\omega_{k}^{2}\tilde{q}_{k}^{2}=(\psi,P_{k})_{2}^{2}+\omega_{k}^{2}(\psi,Q_{k})_{2}^{2}=(p,P_{k})_{2}^{2}+\omega_{k}^{2}(q,Q_{k})_{2}^{2},k=1,\ldots,N$

are the action coordinates of the point $\psi=(q,p)^{T}$. We agree that
$r_{k}(\psi)=\sqrt{r_{k}^{2}(\psi)}\geqslant 0$. It is easy to see that
$r_{k}$ are integrals of the hamiltonian dynamics, that is for any $t\geqslant
0$

$r_{k}^{2}(\psi)=r_{k}^{2}(e^{tA}\psi)$ (8)

The angle variables then are the angles for these oscillators.

The following assertions easily follow from the known facts, see for example
[1], pp. 103, 272. But the proof is very elementary and we give it for the
reader’s convenince in Appendix.

###### Lemma 2

Consider hamiltonian dynamics (3), then

  1. 1.

For any $r_{1}\geqslant 0,\ldots,r_{N}\geqslant 0$ the set

$T(r_{1},\ldots,r_{N})=\\{\psi\in L:r_{k}(\psi)=r_{k},k=1,...,N\\}$

is invariant and diffeomorphic to torus of dimension $N-n$, where $n$ equals
the number of zeros among $r_{1},\ldots,r_{N}$.

  2. 2.

For any point $\psi\in L$ the closure of its orbit coincides with the torus
defined by it, that is

$\overline{\\{e^{tA}\psi:t\geqslant 0\\}}=T(r_{1}(\psi),\ldots,r_{N}(\psi)),$

  3. 3.

Thus the torus defines the vector $\bar{r}=r_{1},r_{2},\ldots,r_{N}$. Vice-
versa, any such vector with non-negative coordinates uniquely defines the
torus. This torus lies on the energy surface $\mathcal{M}_{h}$ iff

$\sum_{k=1}^{N}r_{k}^{2}=2h$

For convenience we put $h=\frac{1}{2}$, and denote the set of all invariant
tori on $\mathcal{M}=\mathcal{M}_{\frac{1}{2}}$ by

$\mathbf{T}=\\{\bar{r}=(r_{1},\ldots,r_{N}):r_{i}\geqslant
0,\sum_{k=1}^{N}r_{k}^{2}=1\\}$

##### Case $N=1$

Here $\omega_{1}^{2}$ can be arbitrary, and we take $h=\omega_{1}^{2}=1$, let
$S$ be the corresponding circle. Then the particle moves along $S$ with
constant angle velocity in the clock-wise direction. Covering theorem is
evident in this case (with $m=1$). However, for $N=1$ we have much stronger
statement.

For any $\psi,\psi^{\prime}$ denote $T(\psi,\psi^{\prime})$ the minimal $t>0$
such that $\psi^{\prime}=e^{At}\psi$. Then $T_{0}=T(\psi,\psi)$ is the first
return time, or the time of complete rotation around the circle, and for any
$\psi$

$T(\psi,-\psi)=\frac{1}{2}T_{0}$

###### Proposition 1

For any $\psi,\psi^{\prime}$ and any $t\geq T_{0}$ there exists
$0<t_{1}=t_{1}(\psi,\psi^{\prime},t)<T_{0}$ such that

$e^{A(t-t_{1})}Ie^{At_{1}}\psi=\psi^{\prime}$

For the proofs see Appendix.

#####  $(r,\tilde{p})$-coordinates and $(r,p)$-coordinates

Note that $I$ can be written as

$I=E-2P_{1},$

where $E$ is the identity matrix and $P_{1}$ is the orthogonal projector on
the vector $g_{1}=(0,e_{1})^{T}\in L$. The expansion

$e_{1}=\sum_{k=1}^{N}\beta_{k}v_{k}$

of the vector $e_{1}\in R^{N}$ defines the numbers $\beta_{k}$. The $p_{k}$
and $\tilde{p}_{k}$ coordinates of the vector $p$ in $R^{N}$ are related by
the formulas

$p=\sum_{i=1}^{N}p_{i}e_{i}=\sum_{k=1}^{N}\tilde{p}_{k}v_{k}$
$\tilde{p}_{k}=\sum_{i=1}^{N}p_{i}(e_{i},v_{k})_{2},\,\,\,p_{k}=\sum_{i=1}^{N}\tilde{p}_{i}(v_{i},e_{k})_{2}$
(9)

Note also that $I$ acts only on $p_{k}$-coordinates and
$\tilde{p}_{k}$-coordinates, but not on $q_{k}$ and $\tilde{q_{k}}$. Thus the
following notation is justified

$Ip=p-2(p,e_{1})_{2}e_{1}=\sum_{k=1}^{N}\tilde{p}^{\prime}_{k}v_{k},\,\,\,\tilde{p}^{\prime}_{k}=\tilde{p}_{k}-2p_{1}\beta_{k}$
(10)

where

$p_{1}=(p,e_{1})_{2}=\sum_{k=1}^{N}\tilde{p}_{k}\beta_{k}.$

By (9) the velocity flip changes the action variables as follows

$r_{k}^{2}(I\psi)=(Ip,v_{k})_{2}^{2}+\omega_{k}^{2}(q,v_{k})_{2}^{2}=(\tilde{p}^{\prime}_{k})^{2}+\omega_{k}^{2}(q,v_{k})_{2}^{2}=\tilde{p}_{k}^{2}+4p_{1}^{2}\beta_{k}^{2}-4\tilde{p}_{k}p_{1}\beta_{k}+\omega_{k}^{2}(q,v_{k})_{2}^{2}=$
$=r_{k}^{2}(\psi)+4p_{1}^{2}\beta_{k}^{2}-4\tilde{p}_{k}p_{1}\beta_{k}=r{}_{k}^{2}(\psi)+4p_{1}^{2}\beta_{k}^{2}-4p_{1}\beta_{k}(p,v_{k})_{2}$

The following formula defines the time evolution of the tori, in terms of
initial action variable and momenta at time $t-0$. Namely, for any $k$

$r_{k}^{2}(Ie^{tA}\psi)=r_{k}^{2}(I\psi(t-0))=r_{k}^{2}(\psi)+4p_{1}^{2}(t-0)\beta_{k}^{2}-4p_{1}(t-0)\beta_{k}(p(t-0),v_{k})_{2},$
(11)

where $\psi(t-0)=e^{tA}\psi=(q(t-0),p(t-0))^{T}\in L,\
p(t-0)=(p_{1}(t-0),\ldots,p_{N}(t-0))^{T}$. We can rewrite (11) as

$\bar{r}(J(t)\psi)=\Psi(\bar{r}(\psi),p(t-0)),$ (12)

where

$\Psi=(\Psi_{1},\ldots,\Psi_{N}),\Psi_{k}^{2}(\bar{r},p)=r_{k}^{2}+4p_{1}^{2}\beta_{k}^{2}-4p_{1}\beta_{k}(p,v_{k})_{2}$

has the domain of definition

$D(\Psi)=\\{(\bar{r},p):\bar{r}\in\mathbf{T},p\in
T_{p}(\bar{r})\\}\subset\mathbf{T}\times\mathbb{R}^{N}$

where $T_{p}(\bar{r})$ is the projection of the torus $T(\bar{r})$ onto the
space of momenta, that is the set of all $p\in\mathbb{R}^{N}$ such that for
some $q\in\mathbb{R}^{N}$ we have $(q,p)^{T}\in T(\bar{r})$. For any
$\psi=\psi(0)=(q,p)^{T},q=q(0),p=p(0)$ consider the following set of tori

$\mathcal{T}_{1}(\psi)=\overline{\\{\bar{r}(J(t)\psi):t\geqslant
0\\}}=\overline{\\{\Psi(\bar{r}(\psi),p(t-0)):t>0,p(0)=p\\}}\subset\mathbf{T}.$

From continuity of $\Psi$ it follows that $\mathcal{T}_{1}(\psi)$ depends only
on the invariant torus containing $\psi$, that is if
$\bar{r}(\psi)=\bar{r}(\psi^{\prime})$ for two points $\psi,\psi^{\prime}\in
L$, then

$\mathcal{T}_{1}(\psi)=\mathcal{T}_{1}(\psi^{\prime})$ (13)

Then, the following notation is correct

$\mathcal{T}_{1}(\psi)=\mathcal{T}_{1}(\bar{r}(\psi))$ (14)

It is useful to note that by continuity of $\Psi$ for any
$\bar{r}\in\mathbf{T}$ we have

$\mathcal{T}_{1}(\bar{r})=\\{\Psi(\bar{r},p):p\in T_{p}(\bar{r})\\}$

##### Case $N=2$

Here we will only give intuitive arguments why for any initial state $\psi$,
in finite number of jumps, we can enter any torus. From (11) one can get the
following formula for the evolution of the action variables

$r_{1}^{2}(J(t)\psi)-r_{1}^{2}(\psi)=4\beta_{1}\beta_{2}(\tilde{p}_{1}(t-0)\beta_{1}+\tilde{p}_{2}(t-0)\beta_{2})(\tilde{p_{2}}(t-0)\beta_{1}-\tilde{p_{1}}(t-0)\beta_{2}),$

and also

$r_{2}^{2}(J(t)\psi)=1-r_{1}^{2}(J(t)\psi)$

Denote the right-hand side of the first formula by $D(t)$. It appears that for
some $t^{\prime}$ the set $\\{D(t):0\leq t\leq t^{\prime}\\}$ contains an open
interval, and moreover its length has lower bound $\delta>0$ uniform in
$\psi$. Thus the measure of the set of visited tori, after each application of
$I$ enlarges by additive constant. It follows that all tori may be visited for
finite number of $I$ transformations. To see this for general $N$ is more
difficult.

###  3.2 Proof of closure theorem

####  3.2.1 Contraction property

Define the function $\rho$ in $\mathbf{T}$

$\rho(\bar{r},\bar{r}^{\prime})=\sum_{k=1}^{N}|r_{k}^{2}-(r^{\prime}_{k})^{2}|,$

where $\bar{r}=(r_{1},\ldots,r_{N}),\
\bar{r}^{\prime}=(r^{\prime}_{1},\ldots,r^{\prime}_{N})$. Denote
$\bar{r}^{*}=(|\beta_{1}|,\ldots,|\beta_{N}|)\in\mathbf{T}$, thus it is the
invariant torus, containing the point $g_{1}=\sqrt{2}(0,e_{1})^{T}\in L$. Note
that all $\beta_{k}$ are nonzero, that follows from the assumption
$V\in\mathbf{V}^{+}$, see section 7.3. The point $g_{1}$ corresponds to the
configuration of particles where all particle have zero velocity and zero
coordinates, except for the particle $1$.

######  Theorem 5 (contraction theorem)

For any $\bar{r}=(r_{1},\ldots,r_{N})\in\mathbf{T}$ we have the following
contraction bound

$\rho(\mathcal{T}_{1}(\bar{r}),\bar{r}^{*})\leqslant(1-c(\bar{r}))\rho(\bar{r},\bar{r}^{*}),$

where the constant $c(\bar{r})$ is given by

$c(\bar{r})=\frac{1}{\max\\{1,D^{2}(\bar{r})\\}},\quad
D(\bar{r})=\max_{k=1,\ldots,N}\frac{r_{k}}{|\beta_{k}|}-\min_{k=1,\ldots,N}\frac{r_{k}}{|\beta_{k}|}$
(15)

Proof. We will find point $p^{{}^{\prime}}\in T_{p}(\bar{r})$ such that

$\rho(\Psi(\bar{r},p^{{}^{\prime}}),\bar{r}^{*})=(1-c(\bar{r}))\rho(\bar{r},\bar{r}^{*}).$

Note first that in coordinates $\tilde{p}$ the set $T_{p}(\bar{r})$ is the
$N$-dimensional cube with sides $(2r_{1},\ldots,2r{}_{N})$, this follows from
the oscillator representation. In other words, the point
$p=(\tilde{p}_{1},\ldots,\tilde{p}_{N})^{T}\in T_{p}(\bar{r})$ iff for any
$k=1,\ldots,N$

$|\tilde{p_{k}}|\leqslant r_{k},$ (16)

For $k=1,\ldots,N$ denote

$\gamma_{k}=\frac{r_{k}}{|\beta_{k}|}.$

and denote the minimal of them by $\gamma_{n}$ and maximal by $\gamma_{N}$. We
will need the following functions

$f^{\pm}(x)=\frac{1}{2}\left(x\pm\sqrt{x^{2}+c(1-x^{2})}\right)$

of $x\in\mathbb{R}$, where the constant $c$ is defined in the formula (15). As
$c\leqslant 1$, then for all $x$

$x^{2}+c(1-x^{2})\geq 0$

Define the point $p^{{}^{\prime}}\in\mathbb{R}^{N}$ with coordinates
$(\tilde{p}_{1}^{{}^{\prime}},\ldots,\tilde{p}_{N}^{{}^{\prime}})$ in the
basis $v_{1},\ldots,v_{N}$ by the formula

$\tilde{p}_{k}^{{}^{\prime}}=y\beta_{k}-c\frac{\beta_{k}^{2}-r_{k}^{2}}{4y\beta_{k}},\quad
y=f^{+}(\gamma_{n}).$

and find the value of $\Psi(\bar{r},p^{{}^{\prime}})$. As

$p_{1}=(p^{{}^{\prime}},e_{1})_{2}=\sum_{k=1}^{N}\beta_{k}\tilde{p}_{k}^{{}^{\prime}}=y\sum_{k=1}^{N}\beta_{k}^{2}-\frac{c}{4y}\sum_{k=1}^{N}(\beta_{k}^{2}-r_{k}^{2})=y.$

then for any $k=1,\ldots,N$ we have

$\Psi_{k}^{2}=r_{k}^{2}+4p_{1}^{2}\beta_{k}^{2}-4p_{1}\beta_{k}\tilde{p}_{k}^{{}^{\prime}}=r_{k}^{2}+4y^{2}\beta_{k}^{2}-4y\beta_{k}\left(y\beta_{k}-c\frac{\beta_{k}^{2}-r_{k}^{2}}{4y\beta_{k}}\right)=r_{k}^{2}+c(\beta_{k}^{2}-r_{k}^{2}).$

Then we have the distance

$\rho(\Psi(\bar{r},p^{{}^{\prime}}),\bar{r}^{*})=\sum_{k=1}^{N}|\Psi_{k}^{2}-\beta_{k}^{2}|=(1-c)\rho(\bar{r},\bar{r}^{*}).$

###### Lemma 3

$p^{{}^{\prime}}\in T_{p}(\bar{r})$ that is the inequalities (16) hold.

Proof consists of simple calculations and is given in Appendix.

From the contraction theorem only, one cannot prove the closure theorem
because $c=c(\bar{r})$ depends on $\bar{r}$. That is why we need one more
assertion. For any $\bar{r}=(r_{1},\ldots,r_{N})\in\mathbf{T}$ put

$A(\bar{r})=\min_{k=1,\ldots,N}\frac{r_{k}}{|\beta_{k}|},\quad
B(\bar{r})=\max_{k=1,\ldots,N}\frac{r_{k}}{|\beta_{k}|},\quad\Delta(\bar{r})=B^{2}(\bar{r})-A^{2}(\bar{r}).$

The function $\Delta(\bar{r})$ defined on $\mathbf{T}$ also looks like a
distance to the point $\bar{r}^{*}$. Let us explain this in more detail. In
particular, $\Delta(\bar{r})=0$ iff $r_{k}=|\beta_{k}|$ for all
$k=1,\ldots,N$. Moreover, as

$\sum_{k=1}^{N}r_{k}^{2}=1$ (17)

then, if $\Delta(\bar{r})$ is small then the numbers $r_{k}$ are close to
$|\beta_{k}|$. Even more, it gives an upper bound for the distance, more
exactly for any $\bar{r}\in\mathbf{T}$

$\rho(\bar{r},\bar{r}^{*})=\sum_{k}|r_{k}^{2}-\beta_{k}^{2}|=\sum_{k}\beta_{k}^{2}|\frac{r_{k}^{2}}{\beta_{k}^{2}}-1|\leqslant\sum_{k}\beta_{k}^{2}(B^{2}(\bar{r})-A^{2}(\bar{r}))=\sum_{k}\beta_{k}^{2}\Delta(\bar{r})\leqslant\Delta(\bar{r})$
(18)

The following result shows that each velocity flip at appropriate moments
makes point $g_{1}$ closer by 1.

###### Corollary 1

For any point $\bar{r}\in\mathbf{T}$ there exists $p\in T_{p}(\bar{r})$ such
that

$\Delta(\Psi(\bar{r},p))\leqslant\max\\{\Delta(\bar{r})-1,0\\}$

Proof. From the proof of contraction theorem it follows that there exists
$p^{{}^{\prime}}\in T_{p}(\bar{r})$ such that

$\Psi_{k}^{2}(\bar{r},p^{{}^{\prime}})=r_{k}^{2}+c(\bar{r})(\beta_{k}^{2}-r_{k}^{2}).$

Denote
$\bar{r}^{\prime}=(r^{\prime}_{1},\ldots,r^{\prime}_{N})=\Psi(\bar{r},p^{{}^{\prime}})$.
Without loss of generality one can choose the indices so that

$\frac{r^{\prime}_{1}}{|\beta_{1}|}=A(\bar{r}^{\prime}),\quad\frac{r^{\prime}_{N}}{|\beta_{N}|}=B(\bar{r}^{\prime}).$

Then

$\Delta(\bar{r}^{\prime})=\frac{(r^{\prime}_{N})^{2}}{\beta_{N}^{2}}-\frac{(r^{\prime}_{1})^{2}}{\beta_{1}^{2}}=(1-c(\bar{r}))\left(\frac{r_{N}^{2}}{\beta_{N}^{2}}-\frac{r_{1}^{2}}{\beta_{1}^{2}}\right).$

from where we get

$\Delta(\bar{r}^{\prime})\leqslant(1-c(\bar{r}))\Delta(\bar{r}).$ (19)

$c(\bar{r})=\frac{1}{\max\\{1,(B(\bar{r})-A(\bar{r}))^{2}\\}}.$

Note that the following inequality holds

$(B-A)^{2}\leqslant B^{2}-A^{2}=\Delta.$

and then

$c(\bar{r})=\frac{1}{\max\\{1,(B(\bar{r})-A(\bar{r}))^{2}\\}}\geqslant\frac{1}{\max\\{1,B^{2}(\bar{r})-A^{2}(\bar{r})\\}}=\frac{1}{\max\\{1,\Delta(\bar{r}\\}}.$

From this inequality and the bound (19) finally we have

$\Delta(\bar{r}^{\prime})\leqslant\left(1-\frac{1}{\max\\{1,\Delta(\bar{r})\\}}\right)\Delta(\bar{r})=\max\\{\Delta(\bar{r})-1,0\\}.$

$\blacktriangledown$

####  3.2.2 Closure theorem

Before proving it we have to prove even weaker assertion.

###### Lemma 4

There exists integer $m\geqslant 1$ such that for any $\psi\in\mathcal{M}$

$g_{1}\in\overline{\mathcal{J}_{m}(\psi)}$

In other words there exists $m$ such that for any $\psi\in\mathcal{M}$ and any
$\epsilon>0$ one can find moments $0<t_{1}<\ldots<t_{m}$ so that

$||J(\tau_{m})\ldots J(\tau_{1})\psi-g_{1}||_{2}<\epsilon,$

By continuity of maps $\Psi$ and $\Delta$, and by Corollary 1 it follows that
for any point $\psi\in L$ and any $\epsilon>0$ one can find time moment
$t\geqslant 0$ such that

$\Delta(\bar{r}(J(t)\psi))=\Delta(\Psi(\bar{r}(\psi),p(t-0)))\leqslant\max\\{\Delta(\bar{r}(\psi))-1,0\\}+\epsilon.$

It follows that there exists $m\geqslant 1$ such that for any $\epsilon>0$
there exist time moments $t_{1},\ldots,t_{m}\geqslant 0$ such that

$\Delta(\bar{r}(\psi^{\prime}))\leqslant\epsilon,\quad\psi^{\prime}=J(\tau_{m})\ldots
J(\tau_{1})\psi.$

and moreover one can take $m=[\Delta(\bar{r}(\psi))]+1$. But as for any
$\bar{r}\in\mathbf{T}$ the following inequality holds

$\Delta(\bar{r})\leqslant\max_{k=1,\ldots,N}\frac{1}{\beta_{k}^{2}},$

$m$ can be chosen uniformly in $\psi$. By formula (18) we get

$\rho(\bar{r}(\psi^{\prime}),\bar{r}^{*})\leqslant\epsilon.$

Alternatively it is evident that there exists constant $c>0$ such that for any
$\epsilon^{\prime}>0$ there is $t\geqslant 0$ such that

$||e^{tA}\psi^{\prime}-g_{1}||_{2}^{2}\leqslant
c\rho(\bar{r}(\psi^{\prime}),\bar{r}^{*})+\epsilon^{\prime}.$

As $\epsilon$ and $\epsilon^{\prime}$ are arbitrary and by $e^{tA}=J(0)J(t)$
we get the proof. $\blacktriangledown$

Now we prove the closure theorem. Define another norm on $L$

$||\psi||_{H}=\sqrt{H(\psi)}$

Let us fix two points $\psi,\psi^{\prime}\in\mathcal{M}$ and show that
$\psi^{\prime}\in\overline{\mathcal{J}_{m}(\psi)}$ for some $m\geqslant 1$ not
depending on $\psi,\psi^{\prime}$. By lemma 4 there exist
$\tau_{1},...,\tau_{m}>0$ and $\tau^{\prime}_{1},...,\tau^{\prime}_{m}>0$ such
that

$||J(\tau_{1},\ldots,\tau_{m})\psi-J(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})\psi^{\prime}||_{H}<\epsilon,$

It is clear that the transform $J(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})$
is invertible and conserves the norm $||\ ||_{H}$, that is why

$||J^{-1}(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})J(\tau_{1},\ldots,\tau_{m})\psi-\psi^{\prime}||_{H}<\epsilon.$

One can find $m^{\prime}\leqslant 2m$ and
$\tau^{\prime\prime}_{1},...,\tau^{\prime\prime}_{m^{\prime}}>0$ such that

$||J(\tau^{\prime\prime}_{1},\ldots,\tau^{\prime\prime}_{m^{\prime}})J(\tau_{1},\ldots,\tau_{m})\psi-\psi^{\prime}||_{H}<2\epsilon,$
(20)

In fact

$J^{-1}(\tau_{1},\ldots,\tau_{m})=e^{-\tau_{1}A}Ie^{-\tau_{2}A}I\ldots
e^{-\tau_{m}A}I$ (21)

As $e^{tA}$ conserves the norm $||.||_{H}$ and $I=J(0)$, it is sufficient to
show that for any point $\psi\in\mathcal{M}$, any $\epsilon>0$ and $t\geqslant
0$ there exists $s=s(\psi,t,\epsilon)$ such that

$||e^{-tA}\psi-e^{sA}\psi||_{H}<\epsilon$

But as the closure of the orbit of any point is the torus, it is Poincare
recurrence theorem.

From (20) and equivalence of the norms $||\ ||_{2}$ and $||\ ||_{H}$ the
closure theorem follows.

###  3.3 Proof of local covering theorem

Fix $\psi$. It is sufficient to show that for some point
$\bar{\tau}\in\Omega_{k}$ the rank of $dJ_{k}^{\psi}(\bar{\tau})$ equals $k$.
The columns of the Jacobian matrix $J_{k}^{\psi}$ are the vectors

$\displaystyle\theta_{k}(\tau)=\frac{d}{d\tau_{k}}J(\tau_{1},\ldots,\tau_{k})\psi=$
$\displaystyle\ Ie^{\tau_{k}A}AJ(\tau_{1},\ldots,\tau_{k-1})\psi$
$\displaystyle\theta_{i}(\tau)=\frac{d}{d\tau_{i}}J(\tau_{1},\ldots,\tau_{k})\psi=$
$\displaystyle\
Ie^{\tau_{k}A}\frac{d}{d\tau_{i}}J(\tau_{1},\ldots,\tau_{k-1})\psi,\quad
i=1,\ldots,k-1.$

Denote $\dim\langle w_{1},w_{2},...\rangle$ the dimension of the linear span
of vectors $w_{1},w_{2},...$. As the mapping $Ie^{\tau_{k}A}$ is non-
degenerate, then for any $k=1,2,...$ one should derive that there is
$\bar{\tau}=(\tau_{1},\ldots,\tau_{k})\in\Omega_{k}$ such that

$\dim\langle\theta_{1}(\tau),\ldots,\theta_{k}(\tau)\rangle=\dim\langle
AJ(\bar{\tau}_{c})\psi,\frac{d}{d\tau_{1}}J(\bar{\tau}_{c})\psi,\ldots,\frac{d}{d\tau_{k-1}}J(\bar{\tau}_{c})\psi\rangle=k,$

where $\bar{\tau}_{c}=(\tau_{1},\ldots,\tau_{k-1})\in\Omega_{k-1}$. We shall
do it by induction in $k$. Remind that for any $\psi\in L$

$I\psi=\psi-2(\psi,g_{1})_{2}g_{1}.$

In fact, for $k=1$ this is trivial.

Induction step. Assuming that we have proved the assertion for $k\leq 2N-2$,
we shall prove it for $k+1$.

For $\bar{\tau}=(\tau_{1},\ldots,\tau_{k})\in\Omega_{k}$ denote
$\bar{\tau}_{c}=(\tau_{1},\ldots,\tau_{k-1})\in\Omega_{k-1}$. Then

$\displaystyle
AJ(\bar{\tau})\psi=AIe^{\tau_{k}A}J(\tau_{1},\ldots,\tau_{k-1})\psi=Ae^{\tau_{k}A}J(\tau_{1},\ldots,\tau_{k-1})\psi+a_{0}(\bar{\tau})Ag_{1}=$
$\displaystyle=e^{\tau_{k}A}\left(AJ(\tau_{1},\ldots,\tau_{k-1})\psi+a_{0}(\bar{\tau})Ae^{-\tau_{k}A}g_{1}\right)=e^{\tau_{k}A}\left(AJ(\bar{\tau}_{c})\psi+a_{0}(\bar{\tau})Ae^{-\tau_{k}A}g_{1}\right),$

where $a_{0}(\bar{\tau})=-2(e^{\tau_{k}A}J(\bar{\tau}_{c})\psi,g_{1})_{2}$.
Similarly we get equalities for the derivatives for $i=1,\ldots,k-1$:

$\displaystyle\frac{d}{d\tau_{i}}J(\bar{\tau})\psi=Ie^{\tau_{k}A}\frac{d}{d\tau_{i}}J(\tau_{1},\ldots,\tau_{k-1})\psi=e^{\tau_{k}A}\frac{d}{d\tau_{i}}J(\tau_{1},\ldots,\tau_{k-1})\psi+a_{i}(\bar{\tau})g_{1}=$
$\displaystyle=e^{\tau_{k}A}\left(\frac{d}{d\tau_{i}}J(\bar{\tau}_{c})\psi+a_{i}(\bar{\tau})e^{-\tau_{k}A}g_{1}\right),\quad
a_{i}(\bar{\tau})=-2(e^{\tau_{k}A}\frac{d}{d\tau_{i}}J(\bar{\tau}_{c})\psi,g_{1})_{2}.$

Derivative in $\tau_{k}$ has the following expansion

$\displaystyle\frac{d}{d\tau_{k}}J(\bar{\tau})\psi=IAe^{\tau_{k}A}J(\tau_{1},\ldots,\tau_{k-1})\psi=Ae^{\tau_{k}A}J(\tau_{1},\ldots,\tau_{k-1})\psi+a_{k}(\bar{\tau})g_{1}=$
$\displaystyle=e^{\tau_{k}A}\left(AJ(\bar{\tau}_{c})\psi+a_{k}(\bar{\tau})e^{-\tau_{k}A}g_{1}\right),\quad
a_{k}(\bar{\tau})=-2(Ae^{\tau_{k}A}J(\bar{\tau}_{c})\psi,g_{1})_{2}.$

Then by nondegeneracy of the mapping $e^{\tau_{k}}A$, we get that the
dimension of subspace

$\langle
AJ(\bar{\tau})\psi,\frac{d}{d\tau_{1}}J(\bar{\tau})\psi,\ldots,\frac{d}{d\tau_{k}}J(\bar{\tau})\psi\rangle$

equals the dimension $\aleph$ of the linear span of $(k+1)$ following vectors

$\displaystyle
AJ(\bar{\tau}_{c})\psi+a_{0}(\bar{\tau})Ae^{-\tau_{k}A}g_{1},AJ(\bar{\tau}_{c})\psi+a_{k}(\bar{\tau})e^{-\tau_{k}A}g_{1},$
$\displaystyle\frac{d}{d\tau_{1}}J(\bar{\tau}_{c})\psi+a_{1}(\bar{\tau})e^{-\tau_{k}A}g_{1},\ldots,\frac{d}{d\tau_{k-1}}J(\bar{\tau}_{c})\psi+a_{k-1}(\bar{\tau})e^{-\tau_{k}A}g_{1}.$

By inductive assumption there exists point $\bar{\tau}_{c}$ such that the
vectors

$AJ(\bar{\tau}_{c})\psi,\frac{d}{d\tau_{1}}J(\bar{\tau}_{c})\psi,\ldots,\frac{d}{d\tau_{k-1}}J(\bar{\tau}_{c})\psi$

are linearly independent. By lemmas 5 and 6 there exists $\tau_{k}\geqslant 0$
such that the vectors

$AJ(\bar{\tau}_{c})\psi,\frac{d}{d\tau_{1}}J(\bar{\tau}_{c})\psi,\ldots,\frac{d}{d\tau_{k-1}}J(\bar{\tau}_{c})\psi,Ae^{-\tau_{k}A}g_{1},e^{-\tau_{k}A}g_{1}$

are linearly independent and
$a_{0}(\bar{\tau})=-2(e^{\tau_{k}A}J(\bar{\tau}_{c})\psi,g_{1})_{2}\neq 0$.
Thus $\aleph=k+1$ and induction step is proved. $\blacktriangledown$

###### Lemma 5

For any vector $\psi\in L\setminus\\{0\\}$ the set

$\\{t>0:\ (e^{tA}\psi,g_{1})_{2}\neq 0\\}\subset\mathbb{R}_{+}$

is open end everywhere dense subset.

Proof. To prove this note the following equalities

$(e^{tA}\psi,g_{1})_{2}=(e^{tA}\psi,g_{1})_{H}=(\psi,e^{-tA}g_{1})_{H}.$

The second because $A^{\star}=-A$, where $A^{\star}$ be the adjoint operator
to $A$ for the scalar product $(,)_{H}$, that follows from the equality

$(Au,w)_{H}=(Vp,q^{\prime})_{2}-(Vq,p^{\prime})_{2}=-((Vq,p^{\prime})_{2}-(p,Vq^{\prime})_{2})=-(u,Aw)_{H}.$

for $u=(q,p)^{T},\ w=(q^{\prime},p^{\prime})^{T}\in L$. Then we use

$(\psi,e^{-tA}g_{1})_{H}=\sum_{k=0}^{\infty}(-1)^{k}\frac{t^{k}}{k!}(\psi,A^{k}g_{1})_{H}$

as, by Lemma 13, the linear span of vectors vectors $A^{k}g_{1},k=0,1,2,...$
coincides with $L$. Then by analiticity of the left-hand part of the latter
formula we get the proof. $\blacktriangledown$

###### Lemma 6

For any linearly independent vectors $w_{1},\ldots,w_{k}\in L,\ k<2N-1$ the
set

$T(w_{1},\ldots,w_{k})=\\{t>0:\dim\langle
e^{-tA}g_{1},Ae^{-tA}g_{1},w_{1},\ldots,w_{k}\rangle=k+2\\}\subset\mathbb{R}_{+}$

contains an open and everywhere dense subset.

Proof. If $k<2N-2$, then choose vectors $w_{k+1},\ldots,w_{2N-2}$ so that
$\dim\langle w_{1},\ldots,w_{2N-2}\rangle=2N-2$. Note that

$T(w_{1},\ldots,w_{2N-2})\subset T(w_{1},\ldots,w_{k})$

That is why it is sufficient to prove the assertion of the lemma for the case
$k=2N-2$. Thus firther on we assume that $k=2N-2$.

Consider the following real function on $L$:

$W(u)=\det[u,Au,w_{1},\ldots,w_{2N-2}].$

Denote $X=\\{u\in L:\ W(u)=0\\}$ the set of zeros of the function $W$. It is
clear that

$T(w_{1},\ldots,w_{2N-2})=\\{t>0:W(e^{-tA}g_{1})\neq 0\\}=\\{t>0:\
e^{-tA}g_{1}\notin X\\}$

It is clear that $W$ is a quadratic function. Thus $X$ is a quadric. Further
we define the canonical type of the quadric X. Denote

$L_{1}=\langle w{}_{1},\ldots,w_{2N-2}\rangle.$

Consider also the orthogonal complement $L_{1}^{\perp}$ to $L_{1}$ (in the
scalar product $(,)_{H}$). By definition of $W$, we have $L_{1}\subset X$.
Thus in some coordinates $(u_{1},u_{2},\ldots,u_{2N})$ on $L$ the function $W$
will look like

$W(u)=a_{1}u_{1}^{2}+a_{2}u_{2}^{2},$

for some $a_{1},a_{2}\in\mathbb{R}$ (as the basis for such coordinates one
could choose the vectors $w_{1},\ldots,w_{2N-2},u_{1},u_{2}$, where
$u_{1},u_{2}$ are the approriate coordinates of the orthogonal compliment
$L_{1}^{\perp}$. As $L_{1}\subset X$, then the form will not depend on the
first $2N-2$ coordinates, and one can choose two coordinates in
$L_{1}^{\perp}$ by method of Lagrange).

If we show that there is exist a vector $u$ such that $W(u)\neq 0$, than it
will follows that $a_{1}$ and $a_{2}$ cannot be simultaneously zero. And
consequently the canonical type of $X$ should be one of three types: a
$2N-2$-dimensional hyperplane, $2N-1$-dimensional hyperplane, the union of two
$2N-1$ hyperplanes. Then applying the same argument as in lemma 5 we get the
proof of lemma 6.

Let show that that there is exist a vector $u$ such that $W(u)\neq 0$.

For vector $u\in L$ denote $u^{\perp},(Au)^{\perp}$ the orthogonal projections
on $L_{1}^{\perp}$ of the vectors $u,Au$ correspondingly. As the determinant
is polilinear we have

$W(u)=\det[u^{\perp},(Au)^{\perp},w_{1},\ldots,w_{2N-2}].$

That is why $X$ is the set of all $u\in L$ such that $u^{\perp}$and
$(Au)^{\perp}$ are linearly dependent. Note that for any $u=(q,p)^{T}\in L$

$(u,Au)_{H}=(Vq,p)_{2}-(p,Vq)_{2}=0.$

Consider two cases:

  1. 1.

Subspace $L_{1}$ is invariant with respect to $A$. Let us show then
$L_{1}^{\perp}$ is also invariant with respect to $A$. In fact, for $u\in
L_{1}^{\perp},\ v\in L_{1}$ we have

$(Au,v)_{H}=(u,A^{\star}v)_{H}=-(u,Av)_{H}=0,$

as $Av\in L_{1}$ by invariance of $L_{1}$. Then $L_{1}^{\perp}$ is invariant
with respect to $A$ and $X\cap L_{1}^{\perp}$ is the set of all $u\in
L_{1}^{\perp}$ such that $u$ and $Au$ are linearly dependent. As the spectrum
of $A$ is pure imaginary and does not contain zero, then for all $u\neq 0\in
L_{1}^{\perp}$ the vectors $u$ and $Au$ are linearly independent. Then

$X\cap L_{1}^{\perp}=\\{0\\}$

and the proof is finished in this case.

  2. 2.

Subspace $L_{1}$ is not invariant with respect to $A$. In this case there
exists vector $v\in L_{1}$ such that $u=(Av)^{\perp}\neq 0$. Consider vector
$\tilde{u}\in L_{1}^{\perp}$ such that $\tilde{u},u$ are linearly independent
and $\tilde{u}^{\prime},u$ are linearly dependent, where
$\tilde{u}^{\prime}\in L_{1}^{\perp}$ is a vector orthogonal to $\tilde{u}$ in
$L_{1}^{\perp}$, again with respect to scalar product $(,)_{H}$. As
$(\tilde{u},A\tilde{u})_{H}=0$, then for the projection $(A\tilde{u})^{\perp}$
of the vector $A\tilde{u}$ on the subspace $L_{1}^{\perp}$ we have

$(A\tilde{u})^{\perp}=c\tilde{u}^{\prime},$

for some $c\in\mathbb{R}$. If $c\neq 0$, then it is clear that
$W(\tilde{u})\neq 0$. Assume that $c=0$ and consider the vector

$u^{*}=v+\tilde{u}.$

Then

$(u^{*})^{\perp}=\tilde{u},\quad(Au^{*})^{\perp}=(Av+A\tilde{u})^{\perp}=u+(A\tilde{u})^{\perp}=u.$

By definition of $\tilde{u}$ the vectors $\tilde{u}$ and $u$ are linearly
independent. Then $(u^{*})^{\perp}$ and $(Au^{*})^{\perp}$ are also linearly
independent, and then $W(u^{*})\neq 0$. The proof in this case is also
finished $\blacktriangledown$

###  3.4 Proof of theorem 1

Fix two points $\psi_{1},\psi_{2}\in\mathcal{M}$. We want to show that for
some $m^{*}$, not depending on $\psi_{1},\psi_{2}$, there exist
$\tau_{1},\ldots,\tau_{m^{*}}\geqslant 0$ such that

$J(\tau_{1},\ldots,\tau_{m^{*}})\psi_{1}=\psi_{2}$

By local covering theorem there exists point
$\bar{\tau}=(\tau_{1},\ldots,\tau_{m})\in\Omega_{m}$ such that there exist
neighborhood $O(\bar{\tau})\subset\Omega_{m}$ of $\bar{\tau}$ and neighborhood
$O(\psi_{1}^{*})\subset\mathcal{M}$ of
$\psi_{1}^{*}=J^{\psi_{1}}(\bar{\tau})=J(\tau_{1},\ldots,\tau_{m})\psi_{1}$
such that

$J^{\psi_{1}}(O(\tau))=O(\psi_{1}^{*}).$ (22)

Consider $\epsilon$-neighborhood $O_{\epsilon}(\psi_{1}^{*})$ of
$\psi_{1}^{*}$ in the norm, corresponding to the scalar product $(,)_{H}$,
such that

$O_{\epsilon}(\psi_{1}^{*})=\\{\psi\in\mathcal{M}:\
||\psi-\psi_{1}^{*}||_{H}<\epsilon\\}\subset O(\psi_{1}^{*})$

By closure theorem there exist
$\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m}\geqslant 0$ such that

$||J(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})\psi_{1}^{*}-\psi_{2}||_{H}<\epsilon.$
(23)

Note that for fixed $\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m}$ the map
$J(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m}):L\to L$ is non-degenerate and
conserves the norm $||\ ||_{H}$. That is why from (23) the following
inequality follows

$||\psi_{1}^{*}-J^{-1}(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})\psi_{2}||_{H}<\epsilon.$

Thus, $J^{-1}(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})\psi_{2}\in
O_{\epsilon}(\psi_{1}^{*})$ and by (22) there is
$\bar{\tau}^{*}=(\tau_{1}^{*},\ldots,\tau_{m}^{*})\in O(\bar{\tau})$ such that

$J(\tau_{1}^{*},\ldots,\tau_{m}^{*})\psi_{1}=J^{-1}(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})\psi_{2}.$

This can be rewritten as

$J(\tau_{1}^{*},\ldots,\tau_{m}^{*},\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})\psi_{1}=J(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m})J(\tau_{1}^{*},\ldots,\tau_{m}^{*})\psi_{1}=\psi_{2}.$

$\blacktriangledown$

##  4 Stochastic part - proof of convergence theorem

###  4.1 Embedded process

Here we consider the sequence $\psi_{k}=\psi(t_{k})$, which is a discrete time
Markov chain (embedded chain) with state space $\mathcal{M}$. We prove that it
is (Markov) ergodic as it is defined in the following theorem concerning more
general class of discrete time Markov chains. Namely, we consider Markov
chains $\xi_{n}$ on compact state space $X$ with Borel $\sigma$-algebra
$\mathcal{B}(X)$ (with countable basis) and transition probability kernels
$P(x,A)$, which are probability measures on $\mathcal{B}(X)$ for any $x\in X$
and measurable functions on $X$ for any $A\subset X$. We will consider the
class of such chains satisfying the following assumptions:

A1) for some integer $m\geq 1$ and any $x\in X$ the $m$-step transition
probability $P^{m}(x,.)$ is equivalent to some finite non-negative measure
$\mu$ such that $\mu(O)>0$ for any open set $O\subset X$. Moreover, for any
$x$ there exists $m$-step transition density $p^{m}(x,y)$ (with respect to
$\mu$), which is measurable on $\mathcal{M}\times\mathcal{M}$;

A2) for any open $O\subset X$ the function $P(x,O)$ is lower semi-continuous.

###### Theorem 6

Under assumptions A1, A2 the Markov chain $\xi_{n}$ is ergodic, that is there
exists probability measure $\pi$ on $X$ such that

$\sup_{A\in\mathcal{B}(X)}|P^{n}(x,A)-\pi(A)|\rightarrow 0$ (24)

as $n\to\infty$, uniformly in $x$.

This theorem follows probably from the existing deep theory of such Markov
chains with continuous state space, see for example [17, 15], but we did not
find the statement we need, and for the reader’s convenience we give a short
proof below, using the ideas from [17, 15].

###### Corollary 2

The embedded chain $\psi_{k}$ has the unique invariant measure $\pi$, and
moreover it is ergodic as in theorem 6.

We have only to prove the properties A1) and A2) for our embedded chain. For
the rest of this section the integer $m$ is the same as in the covering
theorem.

##### Proof of A1)

We will prove that the measures $\pi$ and $P^{m}(\psi,A)$ are equivalent for
any $\psi$.

###### Lemma 7

For any measurable $B\subset\mathcal{M}$ its Liouville measure $\pi(B)=0$ iff
the Lebesgue measure $\lambda$ of the set $(J_{m}^{\psi})^{-1}(B)$ in
$\Omega_{m}$ is zero.

Proof. 1) Assume that for some $B\subset\mathcal{M}$ we have $\pi(B)=0$. Let
us show that $\lambda((J_{m}^{\psi})^{-1}(B))=0$. Let $A_{cr}$ be the set of
critical points of the map $J_{m}^{\psi}$ (that is points
$\overline{\tau}=(\tau_{1},\ldots,\tau_{m})$ where the rank is not maximal)
and let $E=J_{m}^{\psi}(A_{cr})\subset\mathcal{M}$ be the set of critical
values of $J_{m}^{\psi}$. By Sard’s theorem $\pi(E)=0$. But as
$J_{m}^{\psi}(\Omega^{\psi})=\mathcal{M}$, then there exists non-critical
point $\overline{\tau}=(\tau_{1},\ldots,\tau_{m})\in\Omega_{m}$, that is such
that the rank of $dJ_{m}^{\psi}$ at this point equals $2N-1$. As the map
$J_{m}^{\psi}$ is analytic in the variables $\tau_{1},\ldots,\tau_{m}$, the
set of points $A_{cr}$, where the rank is less than $2N-1$, has Lebesgue
measure zero. Then the equality $\lambda((J_{m}^{\psi})^{-1}(B))=0$ follows
from theorem 1 of [13].

2) Assume that for some $B\subset\mathcal{M}$ we have $\pi(B)>0$, and let us
show that $\lambda((J_{m}^{\psi})^{-1}(B))>0$. By Lebesgue differentiation
theorem there exists point $\psi^{\prime}\in\mathcal{M}\setminus E$ and its
neighbourhood $O(\psi^{\prime})$ such that $\pi(O(\psi^{\prime})\cap B)>0$.
Then there is point
$\overline{\tau}=\overline{\tau}(\psi^{\prime})\in(J_{m}^{\psi})^{-1}(\psi^{\prime})$
and some its neighborhood $O(\overline{\tau})\subset\Omega_{m}$, so that the
restriction of $J_{m}^{\psi}$ on $O(\overline{\tau})$ is a submersion. Then
$\pi(O(\psi^{\prime})\cap B)>0$ implies $\lambda((J_{m}^{\psi})^{-1}(B)\cap
O(\overline{\tau}))>0$. $\blacktriangledown$

Denote $\rho^{(m)}$ the product of $m$ densities $\rho$, then as for any
$B\subset\mathcal{M}$

$P^{m}(\psi,B)=\int_{(J_{m}^{\psi})^{-1}(B)}\rho^{(m)}(\overline{\tau})d\overline{\tau}.$

by lemma 7 we get that $P^{m}$ and $\pi$ are equivalent measures.

The proof of measurability of the transition density there is in theorem 1, p.
180 of [16], and in Proposition 1.1, p.5, of [17].

##### Proof of A2)

###### Lemma 8

Our Markov chain $\psi_{k}$ is a weak Feller chain, that is for any open
$O\subset\mathcal{M}$ the transition probability $P(\psi,O)$ is lower
semicontinuous in $\psi$.

Proof. For any $\psi$ denote $\mathbf{1}_{\psi}(\tau)$ the indicator function
on $R_{+}$, that is $\mathbf{1}_{\psi}(\tau)=1$ if $Ie^{\tau A}\psi\in O$, and
zero otherwise. Then we have

$P(\psi,O)=\int_{\mathbb{R}_{+}}\mathbf{1}_{\psi}(\tau)\rho(\tau)d\tau,$

Let $\psi_{n}\rightarrow\psi,\psi_{n}\in\mathcal{M}.$ as $n\rightarrow\infty$.
Fix $\tau\geqslant 0$ and consider two cases:

1\. $Ie^{\tau A}\psi\in O$, then starting from some $n$ the inclusion
$Ie^{\tau A}\psi_{n}\in O$ holds, as $O$ is open. That is why

$\lim_{n\rightarrow\infty}\mathbf{1}_{\psi_{n}}(\tau)=\mathbf{1}_{\psi}(\tau)=1$

2\. $Ie^{\tau A}\psi\notin O$. Then

$\liminf_{n}\mathbf{1}_{\psi_{n}}(\tau)\geqslant\mathbf{1}_{\psi}(\tau)=0.$

Thus for any $\tau$

$\liminf_{n}\mathbf{1}_{\psi_{n}}(\tau)\geqslant\mathbf{1}_{\psi}(\tau)$

Then by Fatou lemma

$\liminf_{n}P(\psi_{n},O)\geqslant P(\psi,O).$

$\blacktriangledown$

###  4.2 Proof of theorem 6

##### Small sets

We will need the following important definition.

###### Definition 7

Below $\nu$ will be any non-zero non-negative measure not necessarily
probabilistic. The Borel subset $C\subset X$ is called $(\nu,n)$-small (or
simply small, if it is $(\nu,n)$-small for some integer $n>0$ and some $\nu$ )
if for any $x\in C$ and any Borel set $B$

$P^{n}(x,B)\geq\nu(B)$

###### Lemma 9

Assume $A1).$ Then for some $n_{1}\geqslant 1$ and some $\nu$ there exists
$(\nu,n_{1})$-small subset $C\in\mathcal{B}(X)$ such that $\nu(C)>0$.

Proof. It follows from the assumption A1) that for any $n\geqslant m$ there is
measurable function $p_{n}(x,y)$ such that

$P^{n}(x,B)=\int_{B}p_{n}(x,y)\mu(dy),$

for any $x\in X,\ B\in\mathcal{B}(X)$. Moreover, $p_{n}(x,y)>0$ for almost any
$(x,y)\in X\times X$ (with respect to $\mu\times\mu$).

We will prove more. Namely, that for some $n\geqslant m$ there exist sets
$B_{1},B_{2}\in\mathcal{B}(X)$ of positive measure $\mu$ and some constant
$\delta>0$ such that for all $x\in B_{1},\ y\in B_{2}$ we have

$p_{n}(x,y)>\delta$ (25)

As the density $p_{m}(x,y)$ is measurable and almost everywhere positive, one
can find number $c>0$ so that the sets

$\displaystyle A=\\{(x,y)\in X\times X:p_{m}(x,y)>c\\}$
$\displaystyle\\{(x,y,z)\in X\times X\times X:\ (x,y)\in A\
\mbox{\leavevmode{\char 232\relax}}\ (y,z)\in A\\}=(A\times X)\cap(X\times A)$

have positive measures $\mu^{2}=\mu\times\mu$ on $X\times X$ and
$\mu^{3}=\mu\times\mu\times\mu$ on $X\times X\times X$ correspondingly. Denote
$O_{r}(x)\subset X$ the open neighborhood of $x\in X$ of radius $r$, and put
$O_{r}(x,y)=O_{r}(x)\times O_{r}(y)\subset X\times X$. By Lebesgue
differentiation theorem there exists a set $A_{0}$ of zero measure
$\mu\times\mu$ such that for any $(x,y)\in A\setminus A_{0}$

$\lim_{r\rightarrow 0+}\frac{\mu^{2}(A\cap
O_{r}(x,y))}{\mu^{2}(O_{r}(x,y))}=1.$

It follows, that the set $((A\setminus A_{0})\times X)\cap(X\times(A\setminus
A_{0}))$ also has positive measure. Consider some point $(x^{*},y^{*},z^{*})$
in this set. Choose $r$ so that the following inequalities hold:

$\displaystyle\mu^{2}(A\cap O_{r}(x^{*},y^{*}))>$ $\displaystyle\
\frac{3}{4}\mu^{2}(O_{r}(x^{*},y^{*}))$ $\displaystyle\mu^{2}(A\cap
O_{r}(y^{*},z^{*}))>$ $\displaystyle\ \frac{3}{4}\mu^{2}(O_{r}(y^{*},z^{*}))$

For any $x\in X$ put

$A_{L}(x)=\\{y\in X:(x,y)\in A\\},\quad A_{R}(z)=\\{y\in X:(y,z)\in A\\}$

Define $B_{1}$ by

$B_{1}=\\{x\in O_{r}(x^{*}):\mu(A_{L}(x)\cap
O_{r}(y^{*}))>\frac{3}{4}\mu(O_{r}(y^{*}))\\}$

Otherwise speaking, the set $B_{1}$ consists of the points $x\in
O_{r}(x^{*})$, for which the set $A_{L}(x)$ is sufficiently large inside
$O_{r}(y^{*})$. Let us show that $B_{1}$ has positive measure. Assume the
contrary \- that for almost all points of $O_{r}(x^{*})$ the inequality
$\mu(A_{L}(x)\cap O_{r}(y^{*}))\leqslant\frac{3}{4}\mu(O_{r}(y^{*}))$ holds.
Then by Fubini theorem

$\mu^{2}(A\cap O_{r}(x^{*},y^{*}))=\int_{O_{r}(x^{*})}\mu(A_{L}(x)\cap
O_{r}(y^{*}))\mu(dx)\leqslant\frac{3}{4}\mu(O_{r}(y^{*}))\mu(O_{r}(x^{*}))=\frac{3}{4}(\mu\times\mu)(O_{r}(x^{*},y^{*})).$

That contradicts the choice of the points $x^{*},y^{*}$. Thus, $\mu(B_{1})>0$.
Similarly, one can show that the set

$B_{2}=\\{z\in O_{r}(z^{*}):\mu(A_{R}(z)\cap
O_{r}(y^{*}))>\frac{3}{4}\mu(O_{r}(y^{*}))\\}$

has positive measure $\mu$. But from the definition of the sets $B_{1},\
B_{2}$ it follows that for any points $x\in B_{1},\ z\in B_{2}$ the following
inequality holds

$\mu(A_{L}(x)\cap A_{R}(z))\geqslant\frac{1}{2}\mu(O_{r}(y^{*})).$

Also we have the estimates for the density $p_{2m}$ and any $x\in B_{1},\ z\in
B_{2}$

$\displaystyle
p_{2m}(x,z)=\int_{X}p_{m}(x,y)p_{m}(y,z)\mu(dy)\geqslant\int_{A_{L}(x)\cap
A_{R}(z)}p_{m}(x,y)p_{m}(y,z)\mu(dy)>$ $\displaystyle>c^{2}\mu(A_{L}(x)\cap
A_{R}(z))\geq\frac{c^{2}}{2}\mu(O_{r}(y^{*})).$

Thus we have proved (25). Now we finish the proof of lemma 9. Note that for
any $x\in B_{1}$ and any $B\in\mathcal{B}(X)$ we have

$P^{n}(x,B)=\int_{B}p_{n}(x,y)\mu(dy)\geqslant\int_{B\cap
B_{2}}p_{n}(x,y)\mu(dy)\geqslant\delta\mu(B\cap B_{2}).$

Moreover, as the sets $B_{1},B_{2}$ have positive measure, then there exists
subset $C\subset B_{2}$ of positive measure $\mu$ and constant
$\delta^{\prime}>0$ such that $P^{m}(x,B_{1})>\delta^{\prime}$ for all $x\in
C$. It follows that for all $x\in C,\ B\in\mathcal{B}(X)$ the following
inequalities hold:

$P^{m+n}(x,B)=\int_{X}P^{m}(x,dy)P^{n}(y,B)\geqslant\int_{B_{1}}P^{m}(x,dy)P^{n}(y,B)\geqslant\delta^{\prime}\delta\mu(B\cap
B_{2}).$

Thus, $C$ is $(\nu,n+m)$-small, where $\nu(B)=\delta\delta^{\prime}\mu(B\cap
B_{2})$, and moreover $\nu(C)>0$. $\blacktriangledown$

###### Lemma 10

Let $C\in\mathcal{B}(X)$ be $(\nu,n_{1})$-small and assume that for any $x$
from some set $D\in\mathcal{B}(X)$

$P^{n}(x,C)>\delta.$

for some $\delta>0$ and $n\geqslant 1$. Then the set $D$ is
$(\delta\nu,n+n_{1})$-small.

Proof. For any $x\in D$ and any $B\in\mathcal{B}(X)$ by semigroup property

$P^{n+n_{1}}(x,B)=\int_{X}P^{n}(x,dy)P^{n_{1}}(y,B)\geqslant\int_{C}P^{n}(x,dy)P^{n_{1}}(y,B)\geqslant
P^{n}(x,C)\nu(B)\geqslant\delta\nu(B),$

$\blacktriangledown$

###### Lemma 11

Under assumptions A1) and A2) the set $X$ itself is small.

Proof. Consider some $(\nu,n_{1})$-small subset $C\in\mathcal{B}(X)$ of lemma
9. For $k=1,2,\ldots$ introduce the subsets

$A_{k}=\\{x\in X:\ P^{m}(x,C)>\frac{1}{k}\\},$

where $m$ is defined in A1). From lemma 10 we have that $A_{k}$ is a
$(\frac{1}{k}\nu,m+n_{1})$-small set. Let us prove that its closure is also a
small set. Note that the measure $\nu$ is regular, that is for any Borel set
$B$

$\nu(B)=\sup\\{\nu(K)\\},$

where the supremum is over all compact sets $K\subset B$. As $\nu(C)>0$, there
exists compact set $K\subset C$ such that for all $x\in A_{k}$

$P^{m+n_{1}}(x,K)\geqslant\frac{1}{k}\nu(K)=\delta>0.$

Let $\\{x_{n}\\}_{n=1,2\ldots}\in A_{k}$ and $x_{n}\rightarrow x$, then by
semi-continuity of the transition probability we get:

$P^{m+n_{1}}(x,K)=1-P^{m+n_{1}}(x,X\setminus K)\geqslant 1-\lim\inf
P^{m+n_{1}}(x_{n},X\setminus K)=\limsup P^{m+n_{1}}(x_{n},K)\geqslant\delta.$

Thus, for any $x\in\bar{A}_{k}$ we have $P^{m+n_{1}}(x,K)>\delta/2$, and
applying lemma 10, we get that $\bar{A}_{k}$ is a
$(\frac{\delta}{2}\nu,n_{2})$-small set for some $n_{2}$.

But also by assumption A1

$X=\cup_{k=1}^{\infty}A_{k}$

As $X$ is not a countable union of sets which are nowhere dense, then for some
$k$ there is an open subset $O\subset\bar{A}_{k}$. Thus for any $x\in X$ and
any $B\in\mathcal{B}(X)$, we have

$P^{m+n_{2}}(x,B)=\int_{X}P^{m}(x,dy)P^{n_{2}}(y,A)\geqslant\int_{O}P^{m}(x,dy)P^{n_{2}}(y,B)\geqslant\frac{\delta}{2}\nu(B)P^{m}(x,O).$
(26)

As the measures $P(x,\cdot)$ and $\mu$ are equivalent for any $x\in X$, we
have $P^{m}(x,O)>0$. But $P^{m}(x,O)$ is lower semi-continuous, and thus
attains minimum on the compact $X$. It follows that

$P^{m}(x,O)>\delta^{\prime}$

for some $\delta^{\prime}>0$ and any $x\in X$. Using inequality (26), we get
the proof.

##### Proof of theorem 6

For$A\in\mathcal{B}(X)$ and $n\geqslant 1$ denote

$I_{n}(A)=\inf_{x\in X}P^{n}(x,A),\quad S_{n}(A)=\sup_{x\in X}P^{n}(x,A).$

Note that

$I_{n+1}(A)=\inf_{x\in X}\int_{X}P(x,dy)P^{n}(y,A)\geqslant\inf_{x\in
X}\int_{X}P(x,dy)I_{n}(A)=I_{n}(A).$

Thus,, for fixed $A\in\mathcal{B}(X)$ the sequence $I_{n}(A)$ is non-
decreasing. Similarly the sequence $S_{n}(A)$ is non-increasing. We shall
prove that $S_{n}(A)-I_{n}(A)$ tends to zero as $n\rightarrow\infty$.

Take number $N$ and measure $\nu$ as in Lemma 11. Then for any $n\geqslant 1$

$P^{n+N}(x,A)=\int_{X}P^{N}(x,dy)P^{n}(y,A)=\int_{X}(P^{N}(x,dy)-\nu(dy))P^{n}(y,A)+\int_{X}\nu(dy)P^{n}(y,A)$

As the measure $P^{N}(x,\cdot)-\nu(\cdot)$ is non-negative, we get from this
equality that

$P^{n+N}(x,A)\geqslant
I_{n}(A)\int_{X}(P^{N}(x,dy)-\nu(dy))+\int_{X}\nu(dy)P^{n}(y,A)=(1-\nu(X))I_{n}(A)+\int_{X}\nu(dy)P^{n}(y,A)$

and then

$I_{n+N}(A)\geqslant(1-\delta)I_{n}(A)+c_{1},\quad\delta=\nu(X),\
c_{1}=\int_{X}\nu(dy)P^{n}(y,A).$

Similarly we get the upper bound for $S_{n+N}(A)$:

$P^{n+N}(x,A)\leq(1-\nu(X))S_{n}(A)+\int\nu(dy)P^{n}(y,A)\leq(1-\delta)S_{n}+c_{1}$

and

$S_{n+N}(A)\leqslant(1-\delta)S_{n}(A)+c_{1}.$

Then the difference is estimated as follows

$S_{n+N}(A)-I_{n+N}(A)\leqslant(1-\delta)(S_{n}(A)-I_{n}(A)).$

From the last inequality and monotonicity of the corresponding sequences it
follows that for any $x\in X$ there exist the following limits and that they
are equal

$\lim_{n\rightarrow\infty}I_{n}(A)=\lim_{n\rightarrow\infty}S_{n}(A)=\lim_{n\rightarrow\infty}P_{n}(x,A)=\pi(A),$

and moreover the convergence is uniform in $A\in\mathcal{B}(X)$ and in $x$.
$\blacktriangledown$

###  4.3 Proof of theorem 2

We will use the following theorem (strong law of large numbers) for discrete
time Markov chains on arbitrary state space $X$ equipped with $\sigma$-algebra
$\mathcal{A}$. Let $P^{n}(x,B)$ be $n$-step transition probability assumed to
be measurable on $X$ for any $B\in\mathcal{A}$ and is a probability measure on
$(X,\mathcal{A})$ for any $x$. Let us assume that there exists invariant
measure $\pi$ on $(X,\mathcal{A})$ such that uniformly in $x$

$\sup_{A\in\mathcal{A}}|P^{n}(x,A)-\pi(A)|\rightarrow 0,\quad
n\rightarrow\infty$

Denote $P_{x}$ the measure on trajectories $(x_{0}=x,x_{1},x_{2},...)$ with
initial point $x$. Under these conditions the following assertion holds.

###### Theorem 8

For any $f\in L^{1}(X,\pi)$ and any $x\in X$ we have $P_{x}$-a.s.

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n}f(x_{k})=\int_{X}f(x)\pi(dx)$

Proof. See [18], p. 140, and [16], p. 209.

To prove theorem 2 we need the following lemma.

###### Lemma 12

For any measurable bounded function $f$ on $\mathcal{M}$ and any initial state
$\psi(0)=\psi$ the following limit holds a.s.

$\lim_{N\rightarrow\infty}M_{f}(t_{N})=\pi(f)$

Proof. Denote $X_{k}=(\psi_{k},\tau_{k+1}),\ k=0,1,\ldots$ the Markov chain
with values in $X=\mathcal{M}\times\mathbb{R}_{+}$. Then

$\int_{t_{k}}^{t_{k+1}}f(\psi(s))ds=\int_{t_{k}}^{t_{k+1}}f(e^{(s-t_{k})A}\psi_{k})ds=\int_{0}^{\tau_{k+1}}f(e^{sA}\psi_{k})ds=F(X_{k})$
(27)

where

$F(\psi,t)=\int_{0}^{t}f(e^{sA}\psi)ds,\ (\psi,t)\in X.$

Then

$M_{f}(t_{N})=\frac{1}{t_{N}}\sum_{k=0}^{N-1}\int_{t_{k}}^{t_{k+1}}f(\psi(s))ds=\frac{1}{t_{N}}\sum_{k=0}^{N-1}F(X_{k})$
(28)

It is easy to show that $X_{k}$ has invariant measure $\mu=\pi\times
P_{\tau},P_{\tau}=\rho ds,$ satisfies the conditions of theorem 8 as
$\psi_{k}$ satisfies it. Then

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=0}^{N-1}F(X_{k})=\mu(F)=\int_{X}F(\psi,s)d\mu$

where

$\mu(F)=\int_{\mathbb{R}_{+}}P_{\tau}(dt)\int_{\mathcal{M}}\pi(d\psi)\int_{0}^{t}dsf(e^{sA}\psi)=\int_{\mathbb{R}_{+}}P_{\tau}(dt)\int_{0}^{t}ds\int_{\mathcal{M}}\pi(d\psi)f(e^{sA}\psi)=$
$=\pi(f)\int_{\mathbb{R}_{+}}P_{\tau}(dt)\int_{0}^{t}ds=\pi(f)E\tau_{1}$

Moreover, by strong law of large numbers for independent random varoables
$\tau_{k}$ we have

$\lim_{N\rightarrow\infty}\frac{t_{N}}{N}=E\tau_{1}$

Then by (28) we get the proof of the lemma.$\blacktriangledown$

To prove theorem 2 we have to estimate the difference between $M_{f}(t)$ and
$M_{f}(t_{N})$. Using the boundedness $|f(\psi)|\leqslant c$ we have

$|M_{f}(t)-M_{f}(t_{N})|\leqslant|\frac{1}{t}\int_{t_{N}}^{t}f(\psi(s))ds|+\frac{|t-t_{N}|}{t}|M_{f}(t_{N})|\leqslant\frac{|t-t_{N}|}{t}(c+|M_{f}(t_{N})|).$
(29)

For any $t>0$ define the random index $N(t)$ so that

$t_{N(t)}\leqslant t<t_{N(t)+1}.$

and note that

$\frac{|t-t_{N(t)}|}{t}\leqslant\frac{\tau_{N(t)+1}}{t_{N(t)}}=\frac{\tau_{N(t)+1}}{\sum_{k=1}^{N(t)}\tau_{k}}.$

As $E\tau_{1}<\infty$, the law of large numbers, as $N\to\infty$, gives a.s.

$\frac{\tau_{N+1}}{\sum_{k=1}^{N}\tau_{k}}\rightarrow 0$

But $N(t)\rightarrow\infty$ as $t\rightarrow\infty$. Then the right-hand side
of (29) tends to $0$ a.s. as $N=N(t)$ and $t\rightarrow\infty$.
$\blacktriangledown$

##  5 Appendix

###  5.1 Proof of lemma 2

For any $t\geqslant 0$ one can show that

$e^{tA}=\left(\begin{array}[]{cc}\cos(t\sqrt{V})&(\sqrt{V})^{-1}\sin(t\sqrt{V})\\\
-\sqrt{V}\sin(t\sqrt{V})&\cos(t\sqrt{V})\end{array}\right),$

where $\sqrt{V}$ is the positive square root of the matrix $V$. Then for any
$k=1,\ldots,N$ and $t\geqslant 0$.

$\displaystyle e^{tA}Q_{k}=$ $\displaystyle\
\cos(\omega_{k}t)Q_{k}-\omega_{k}\sin(\omega_{k}t)P_{k},$ (30) $\displaystyle
e^{tA}P_{k}=$ $\displaystyle\
\frac{\sin(\omega_{k}t)}{\omega_{k}}Q_{k}+\cos(\omega_{k}t)P_{k},$ (31)

Using (30)-(31) we have for any $\psi$

$\psi(t)=e^{tA}\psi=\sum_{k=1}^{N}\left(\cos(\omega_{k}t)\tilde{q}_{k}+\frac{\sin(\omega_{k}t)}{\omega_{k}}\tilde{p}_{k}\right)Q_{k}+\sum_{k=1}^{N}\left(-\omega_{k}\sin(\omega_{k}t)\tilde{q}_{k}+\cos(\omega_{k}t)\tilde{p}_{k}\right)P_{k}.$

and then

$\displaystyle\tilde{q}_{k}(t)=$ $\displaystyle\
\cos(\omega_{k}t)\tilde{q}_{k}+\frac{\sin(\omega_{k}t)}{\omega_{k}}\tilde{p}_{k}$
$\displaystyle\tilde{p}_{k}(t)=$ $\displaystyle\
-\omega_{k}\sin(\omega_{k}t)\tilde{q}_{k}+\cos(\omega_{k}t)\tilde{p}_{k}.$

From these two formulas we see that pair of functions
$(\tilde{q}_{k}(t),\tilde{p}_{k}(t)),k=1,\ldots,N,$ corresponds to the
dynamics of one-dimensional oscillator of unit mass and frequency
$\omega_{k}$, where $\tilde{q}_{k}(t)$ is the oscillator coordinate and
$\tilde{p}_{k}(t)$ is its momentum. Thus the dynamics $e^{tA}\psi$ is
isomorphic to the uniform movement on the torus with velocity
$(\omega_{1}^{2},\ldots,\omega_{N}^{2})$. This gives the second assertion.

###  5.2 Case $N=1$

Note the identity for any $\psi$

$Ie^{At_{1}}\psi=e^{-At_{1}}I\psi$ (32)

Let us first prove the proposition for $t=T_{0}$, $\psi=\psi_{0}=(1,0)$ and
arbitrary $\psi^{\prime}$. Then for any define the one-to-one mapping
$W:(0,\frac{1}{2}T_{0})\to S$. namely for any $t_{1}\in(0,\frac{1}{2}T_{0})$
we define

$W(t^{\prime})=\psi^{\prime}=e^{A(T_{0}-t_{1})}Ie^{At_{1}}\psi_{0}=e^{A(T_{0}-2t_{1})}\psi_{0}$

Then it is sufficient to take $t^{\prime}=\frac{T_{0}-t^{\prime\prime}}{2}$
and choose minimal $t^{\prime\prime}>0$ so that

$\psi^{\prime}=e^{At^{\prime\prime}}\psi_{0}$

For $t>T_{0}$ the proof is quite similar but we will not need this case to
prove convergence.

###  5.3 Mixing subspace

Here we give some properties of the mixing space. The following two lemmas
show how the dimension of $L_{-}$can be explicitely characterized.

###### Lemma 13

The space $L_{-}$ is invariant with respect to $A$. Moreover

$L_{-}=\langle\\{A^{k}g_{1}:\ k=0,1,\ldots\\}\rangle$

where $\langle\ \rangle$ is the linear span of the set of vectors.

Consider also the orthogonal complement to $L_{-}$ in the scalar product
$(,)_{2}$

$L_{0}=L_{-}^{\perp}.$

Then it is also invariant with respect to $A$. Moreover, the vector $\psi\in
L_{0}$ iff for the hamiltonian dynamics with initial condition $\psi=\psi(0)$
the momentum $p_{1}(t)=0$ for any $t$.

###### Lemma 14

Assume that the spectrum of $V$ is simple, and let $\\{v_{1},\ldots,v_{N}\\}$
be the eigenvectors of $V$, they form a basis in $R^{N}$. Then the dimension
of $L_{0}$ is twice the number of $v_{k}$ having coordinates
$v_{k,1}=(e_{1},v_{k})=0$.

What occurs if condition (4) is not fulfilled, exact formulas for the
dimension of $L_{-}$ for the chain of harmonic oscillators and for other cases
see in [11, 10, 9].

###  5.4 Proof of lemma 3

We have

$|\tilde{p}_{k}^{*}|=|\beta_{k}|\left|y-c\frac{1-\gamma_{k}^{2}}{4y}\right|=|\beta_{k}|\left|\frac{2\gamma_{n}^{2}+2\gamma_{n}\sqrt{\gamma_{n}^{2}+c(1-\gamma_{n}^{2})}+c(\gamma_{k}^{2}-\gamma_{n}^{2})}{4y}\right|$

As for any $k=1,\ldots,N$ we have $\gamma_{k}\geqslant\gamma_{n}\geqslant 0$,
then the expression under module in the last formula is non-negative, and we
have

$|\tilde{p}_{k}^{{}^{\prime}}|=|\beta_{k}|\left(y-c\frac{1-\gamma_{k}^{2}}{4y}\right).$

Consider two cases:

  1. 1.

$\gamma_{k}\leqslant 1.$ Show that $f^{+}(x)$ is monotone increasing, that is
its derivative

$(f^{+}(x))^{\prime}=\frac{1}{2}\left(1+\frac{x(1-c)}{\sqrt{x^{2}+c(1-x^{2})}}\right)>0.$

Thus $y=f^{+}(\gamma_{n})\leqslant f^{+}(\gamma_{k})$. Taking into account
$\gamma_{k}\leqslant 1$, we have the inequalities

$\displaystyle|\tilde{p}_{k}^{{}^{\prime}}|=|\beta_{k}|\left(y-c\frac{1-\gamma_{k}^{2}}{4y}\right)\leqslant|\beta_{k}|\left(f^{+}(\gamma_{k})-\frac{c(1-\gamma_{k}^{2})}{4f^{+}(\gamma_{k})}\right)=$
$\displaystyle=|\beta_{k}|\frac{2\gamma_{k}^{2}+2\gamma_{k}\sqrt{\gamma_{k}^{2}+c(1-\gamma_{k}^{2})}}{4f^{+}(\gamma_{k})}=|\beta_{k}|\gamma_{k}=r_{k}.$

  2. 2.

$\gamma_{k}>1$. Then we will show that $f^{-}(\gamma_{k})\leqslant y$. It is
easy to check that $f^{-}(x)$ is increasing and then

$0<f^{-}(\gamma_{k})\leqslant f^{-}(\gamma_{N}).$

Note that the left inequality holds because $f^{-}(1)=0$.

Let us prove that for any $x\in\mathbb{R}$ we have $f^{-}(x)<\frac{c}{2}x$.
Consider the chain of inequalities

$f^{-}(x)-\frac{c}{2}x=\frac{1}{2}\left((1-c)x-\sqrt{x^{2}(1-c)+c}\right)=\frac{1}{2}\left(\frac{-c-x^{2}(1-c)c}{(1-c)x+\sqrt{x^{2}(1-c)+c}}\right)<0$

Thus we have proved that

$f^{-}(\gamma_{k})\leqslant f^{-}(\gamma_{N})<\frac{c}{2}\gamma_{N}.$ (33)

By $f^{+}(0)=\frac{1}{2}\sqrt{c}$ and the evident inequality
$(f^{+}(x))^{\prime}\geqslant\frac{c}{2}$, which holds as $c\leqslant 1$, we
have

$y=f^{+}(\gamma_{n})\geqslant\frac{1}{2}(c\gamma_{n}+\sqrt{c}).$ (34)

Then by (33), (34) and $c\leqslant\frac{1}{(\gamma_{N}-\gamma_{n})^{2}}$ we
get

$y-f^{-}(\gamma_{k})\geqslant\frac{1}{2}(-c(\gamma_{N}-\gamma_{n})+\sqrt{c})=\frac{\sqrt{c}}{2}\left(-\sqrt{c}(\gamma_{N}-\gamma_{n})+1\right)\geqslant
0.$

Remind that we have proved earlier that the function $f^{+}(x)$ is increasing,
then taking into account the latter inequality we have the following bounds

$\displaystyle|\tilde{p}_{k}^{*}|=|\beta_{k}|\left(y-c\frac{1-\gamma_{k}^{2}}{4y}\right)=|\beta_{k}|\left(y+c\frac{\gamma_{k}^{2}-1}{4y}\right)\leqslant|\beta_{k}|\left(f^{+}(\gamma_{k})+c\frac{\gamma_{k}^{2}-1}{4f^{-}(\gamma_{k})}\right)=$
$\displaystyle=|\beta_{k}|\left(\frac{\gamma_{k}^{2}-\gamma_{k}^{2}-c(1-\gamma_{k}^{2})+c(\gamma_{k}^{2}-1)}{4f^{-}(\gamma_{k})}\right)=|\beta_{k}|c\frac{\gamma_{k}^{2}-1}{2f^{-}(\gamma_{k})}=|\beta_{k}|c\frac{2(\gamma_{k}^{2}-1)f^{+}(\gamma_{k})}{2c(\gamma_{k}^{2}-1)}=$
$\displaystyle=|\beta_{k}|f^{+}(\gamma_{k})\leqslant|\beta_{k}|\gamma_{k}=r_{k}.$

In the last inequality we used fact that $f^{+}(x)<x$ for $x>1$ holds due to
$f^{+}+f^{-}=x$. $\blacktriangledown$

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