# Non-uniform ergodic properties of Hamiltonian flows with impacts

Krzysztof Frączek  Faculty of Mathematics and Computer Science, Nicolaus
Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
<EMAIL_ADDRESS>and  Vered Rom-Kedar  The Estrin Family Chair of Computer
Science and Applied Mathematics. Department of Computer Science and Applied
Mathematics, The Weizmann Institute, Rehovot, Israel 76100  Vered.Rom-
<EMAIL_ADDRESS>
###### Abstract.

The ergodic properties of two uncoupled oscillators, a horizontal and vertical
one, residing in a class of non rectangular star-shaped polygons with only
vertical and horizontal boundaries and impacting elastically from its
boundaries are studied. We prove that the iso-energy level sets topology
changes non-trivially; the flow on level sets is always conjugated to a
translation flow on a translation surface, yet, for some segments of partial
energies the genus of the surface is strictly larger than one. When at least
one of the oscillators is un-harmonic, or when both are harmonic and non-
resonant, we prove that for almost all partial energies, including the
impacting ones, the flow on level sets is unique ergodic. When both
oscillators are harmonic and resonant, we prove that there exist intervals of
partial energies on which periodic ribbons and additional ergodic components
co-exist. We prove that for almost all partial energies in such segments the
motion is unique ergodic on the part of the level set that is not occupied by
the periodic ribbons. This implies that ergodic averages project to piecewise
smooth weighted averages in the configuration space.

Research partially supported by the Narodowe Centrum Nauki Grant
2017/27/B/ST1/00078 and by ISF Grant 1208/16.

##  1\. Introduction

Hamiltonian Impact Systems (HIS) describe the motion of a particle in a given
Hamiltonian field within a billiard table: the Hamiltonian flow determines the
particle trajectory in the configuration space till it reaches the billiard
boundary, there it reflects elastically, and then it continues with the
Hamiltonian flow [10]. For mechanical Hamiltonian flows with bounded energy
surfaces, for small energy, as long as the energy surface projection to the
configuration space (the Hill’s region) does not touch the billiard boundary,
the HIS reduces to the study of smooth mechanical Hamiltonian systems. On the
other extreme, at the high energy limit, for compact billiard tables,
mechanical HIS limit to the billiard flow in the specified billiard table. The
theory for intermediate energy values includes local analysis near periodic
orbits [5] and near smooth convex boundaries [22, 2], and, for some specific
classes of HIS, hyperbolic behavior [19], Liuoville integrable [11, 6, 3, 16,
14] and near-integrable [15] dynamics were established. A class of quasi-
integrable HIS, which is related to the quasi-integrable dynamics in families
of polygonal right angled corners, was introduced in [1].

The analysis of quasi-integrable dynamics in right angled billiards is related
to several deep mathematical fields [23]. A new family of billiards with
quasi-integrable dynamics, consisting of confocal ellipses with confocal
barriers was introduced in [4] (and, if the Birkhoff conjecture is correct,
this family and polygonal billiards with rational angles are the only
billiards with quasi-integrable dynamics). For such a billiard table, the
quasi-integrable dynamics depend on a parameter - the constant of motion
associated with the caustic of the trajectories. By a change of coordinates,
the dynamics for any given caustic constant is conjugated to the directed
motion in a right angled billiard table [4]. Using tools of homogeneous
dynamics it was established that the flow in an ellipse with a vertical
barrier is uniquely ergodic for almost all the caustic parameters [9].
Developing a different approach, a similar result was established for the more
general case of nibbled ellipses [8]. Our methodology relies on the methods
developed in [8], where it was shown that to prove unique ergodicity, the
Minsky-Weiss criterion [13] may be applied to a class of right angled polygons
consisting of staircase polygons.

Here, we examine the dynamics of a horizontal and a vertical oscillator with
stable fixed point at the origin that are restricted to lie within star-shaped
polygons with only vertical and horizontal boundaries with a kernel that
includes the origin (this is a subclass of the HIS introduced in [1], and such
polygons consist of $4$ staircase polygons considered in [8]). These $2$
degrees of freedom systems have $2$ conserved integrals, so their motion is
always restricted to level sets, yet, in contrast to the smooth case, the
motion on the level sets is conjugated, for some of the level sets, to the
motion in polygonal right angled billiards with more than $4$ corners. Thus,
the motion on such level sets is not conjugated to rotations [23], and, since
the shape of the polygonal billiard and the direction of motion on it vary,
the dynamics may depend sensitively on the value of the conserved integrals,
even for iso-energy level sets [1]. In the first part of the paper we analyse
our class of HIS in non-resonant cases and prove that the motion is unique-
ergodic for almost all iso-energy level sets (in this part we rely on the
tools and analysis developed in [8] for staircase polygons, verifying that the
functional dependence of the corners in the induced family of polygons
satisfies the needed conditions of smoothness, independence and monotonicity
as in [8]). In the second part of the paper we study the case of HIS with
quadratic resonant potentials in a staircase polygon. Here, ribbons of
periodic orbits co-exist with quasi-periodic motion (similar to the motion
presented in [12], but in the corresponding HIS). By our new construction, we
establish the unique ergodicity of the non-periodic component on a certain
set. In Section 9 we show that these findings imply non-uniform ergodic
averages in the configuration space.

###  1.1. Set up

Consider integrable 2 d.o.f. (degrees of freedom) Hamiltonian systems of the
form

(1.1)
$H(p_{x},x,p_{y},y)=\frac{p_{x}^{2}}{2}+\frac{p_{y}^{2}}{2}+V_{1}(x)+V_{2}(y),$

where $(x,y)$ are the space coordinates, $(p_{x},p_{y})$ are the corresponding
momenta and the potentials $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$
are even uni-modal $C^{2}$-maps that tend monotonically to infinity with their
argument (with no loss of generality we take $V_{1}(0)=V_{2}(0)=0$, see
precise assumption "Deck" below). The Hamiltonian flow
$(\varphi_{t})_{t\in{\mathbb{R}}}$ of (1.1) describes a particle which
oscillates in a potential well. The Hamiltonian flow on a given energy surface
$E$ is foliated by the level sets with fixed partial energies,
$(E_{1}=H_{1}(I_{1}),E_{2}=H_{2}(I_{2})=E-E_{1})$, where $I_{i}(E_{i})$ is the
action of the one d.o.f. system $H_{i}$. For a given energy level $E\geq 0$
and any $0\leq E_{1}\leq E$ let

$S_{E,E_{1}}:=\Big{\\{}(p_{x},x,p_{y},y)\in{\mathbb{R}}^{4}:\frac{p_{x}^{2}}{2}+V_{1}(x)=E_{1},\frac{p_{y}^{2}}{2}+V_{2}(y)=E-E_{1},(x,y)\in{\mathbb{R}}^{2}\Big{\\}}.$

Then the phase space of the flow $(\varphi_{t})_{t\in{\mathbb{R}}}$, is
foliated by the invariant sets $\\{S_{E,E_{1}}:E\geq 0,0\leq E_{1}\leq E\\}$,
which are tori for $0<E_{1}<E,$ and, for $E>0$ and $E_{1}\in\\{0,E\\}$, are
circles. Denote the restriction of $(\varphi_{t})_{t\in{\mathbb{R}}}$ to
$S_{E,E_{1}}$ by $(\varphi^{E,E_{1}}_{t})_{t\in{\mathbb{R}}}$. The smooth flow
without reflection is trivially integrable and oscillatory. The projection of
$S_{E,E_{1}}$ to the configuration space is the projected rectangle [14]:

(1.2)
$R^{(E,E_{1})}=[-x^{max}(E_{1}),x^{max}(E_{1})]\times[-y^{max}(E-E_{1}),y^{max}(E-E_{1})],$

where $V_{1}(x^{max}(E_{1}))=E_{1},V_{2}(y^{max}(E_{2}))=E_{2},E_{2}=E-E_{1}$.
The union of all iso-energy rectangles is the Hill region:
$\mathcal{D}_{Hill}(E)=\bigcup_{0\leq E_{1}\leq
E}R^{(E,E_{1})}=\\{(x,y)|V_{1}(x)+V_{2}(y)\leqslant E\\}$ (see [14] for more
general formulation).

Denote by $\omega_{i}(E_{i})=\frac{2\pi}{T_{i}(E_{i})}$ the frequency in each
degree of freedom, where $T_{i}(E_{i})$ is the period of oscillation. The
standard transformation to action angle coordinates $(I_{i},\theta_{i})$ in
each degree of freedom brings (1.1) to the form
$H(p_{x},x,p_{y},y)=H_{1}(I_{1})+H_{2}(I_{2})$ and in these coordinates the
flow is simply
$\varphi^{E,E_{1}}_{t}=\\{I_{1}(E_{1}),\theta_{1}(t)=\omega_{1}(E_{1})t+\theta_{1}(0),I_{2}(E-E_{1}),\theta_{2}(t)=\omega_{2}(E-E_{1})t+\theta_{2}(0)).$
Recall that $\omega_{i}(I_{i})=H_{i}^{\prime}(I_{i})=\frac{dE_{i}}{dI_{i}}$
and that the Hamiltonian is said to satisfy the twist condition if
$det(\frac{\partial^{2}H}{\partial I_{i}\partial
I_{j}})=\prod\omega_{i}^{\prime}(I_{i})\neq 0$ and to satisfy the iso-energy
non-degeneracy condition if

$\begin{vmatrix}\frac{\partial^{2}H}{\partial I_{i}\partial
I_{j}}&\frac{\partial H}{\partial I_{i}}\\\ \frac{\partial H}{\partial
I_{j}}&0\\\
\end{vmatrix}=\begin{vmatrix}\omega_{1}^{\prime}(I_{1})&0&\omega_{1}(I_{1})\\\
0&\omega_{2}^{\prime}(I_{2})&\omega_{2}(I_{2})\\\
\omega_{1}(I_{1})&\omega_{2}(I_{2})&0\\\
\end{vmatrix}=-\omega_{1}^{\prime}(I_{1})\omega_{2}^{2}(I_{2})-\omega_{1}^{2}(I_{1})\omega_{2}^{\prime}(I_{2})\neq
0.$

The character of the smooth flow $\varphi^{E,E_{1}}_{t}$ on the level set
$S_{E,E_{1}}$ depends on the frequency ratio on this level set. If

$\Omega=\Omega(E,E_{1})=\frac{\omega_{1}(E_{1})}{\omega_{2}(E-E_{1})}=\frac{H^{\prime}_{1}(I_{1}(E_{1}))}{H^{\prime}_{2}(I_{2}(E-E_{1}))}$

is rational the flow is periodic (the resonant case) and if $\Omega$ is
irrational it is quasi-periodic (the non-resonant case). Recall that
$\frac{d}{dE_{1}}\Omega(E,E_{1})\neq 0$ iff the iso-energy non-degeneracy
condition is satisfied.

Now, assume that the particle is confined to a bounded polygonal room
$P\subset{\mathbb{R}}^{2}$ whose walls consist of vertical and horizontal
segments. When the particle meets the wall it reflects elastically. More
precisely, if a trajectory meets a vertical segment at $(p_{x},x,p_{y},y)$
then it jumps to $(-p_{x},x,p_{y},y)$ and continues its movement in accordance
with the Hamiltonian flow solving:

(1.3)
$\frac{dp_{x}}{dt}=-V_{1}^{\prime}(x),\quad\frac{dx}{dt}=p_{x},\quad\frac{dp_{y}}{dt}=-V_{2}^{\prime}(y),\quad\frac{dy}{dt}=p_{y}.$

Similarly, if a trajectory meets a horizontal segment at $(p_{x},x,p_{y},y)$
then it jumps to $(p_{x},x,-p_{y},y)$ and continues its movement with (1.3),
see [1] for the general construction, a mechanical example and the description
of the resulting dynamics on energy surfaces and [14] for global structure of
energy surfaces of such systems.

In particular, since all the walls are either horizontal or vertical, the
partial energies are preserved under these reflections, so the motion remains
restricted to level sets:

$S^{P}_{E,E_{1}}:=\\{(p_{x},x,p_{y},y)\in{\mathbb{R}}^{4}:H_{1}(x,p_{x})=E_{1},H_{2}(y,p_{y})=E-E_{1},(x,y)\in
P\Big{\\}}.$

Denote the restriction of the impact Hamiltonian flow
$(\varphi_{t})_{t\in{\mathbb{R}}}$ to $S^{P}_{E,E_{1}}$ by
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$, see for example a trajectory
segment projected to the configuration space in Figure 2. Notice that if a
particle hits any corner of $P$ then it dies. Namely, the flow is fully
defined for regular trajectories and is only partially defined on the set
which corresponds to all forward and backward images of corner points.

We are interested in studying the topology of the level set $S^{P}_{E,E_{1}}$
and the invariant measures of $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$
when the total and partial energies $E,E_{1}$ vary. In particular, we ask when
the flow is uniquely ergodic. Recall that the flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic if (i) each
of its orbits is forward or backward infinite and (ii) if there exists a
probability measure $\mu_{E,E_{1}}$ on $S^{P}_{E,E_{1}}$ such that for every
continuous map $f:S^{P}_{E,E_{1}}\to{\mathbb{R}}$ and any
$(p_{x},x,p_{y},y)\in S^{P}_{E,E_{1}}$ for which the
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$-orbit of $(p_{x},x,p_{y},y)$ is
either forward or backward infinite

$\lim_{T\to\pm\infty}\frac{1}{T}\int_{0}^{T}f(\varphi_{t}(p_{x},x,p_{y},y))dt=\int_{S^{P}_{E,E_{1}}}f\,d\mu_{E,E_{1}},$

where the $+$ (respectively $-$) sign is taken if the
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$-orbit of $(p_{x},x,p_{y},y)$ is
forward (respectively backward) infinite. In our case the measure
$\mu_{E,E_{1}}$ is equivalent to the Lebesgue measure on $S^{P}_{E,E_{1}}$.

To formally determine the living space, $P$, of the particle, by following
[8], denote by $\Xi$ the set of sequences
$(\overline{x},\overline{y})=(x_{i},y_{i})_{i=1}^{k}$ of points in
${\mathbb{R}}^{2}_{>0}$ such that

$0<x_{1}<x_{2}<\ldots<x_{k-1}<x_{k}\quad\text{and}\quad
0<y_{k}<y_{k-1}<\ldots<y_{2}<y_{1}.$

Figure 1. Staircase polygons $P(\overline{x},\overline{y})$,
$P(-\overline{x},\overline{y})$, $P(\overline{x},-\overline{y})$,
$P(-\overline{x},-\overline{y})$.

For every $(\overline{x},\overline{y})\in\Xi$ set
$k(\overline{x},\overline{y}):=k$ and denote by $P(\overline{x},\overline{y})$
the right-angle staircase polygon on ${\mathbb{R}}^{2}$ with consecutive
vertices:

$(0,0),(0,y_{1}),(x_{1},y_{1}),(x_{1},y_{2}),\ldots,(x_{k-1},y_{k-1}),(x_{k-1},y_{k}),(x_{k},y_{k}),(x_{k},0),$

see Figure 1.

Denote by $\Gamma$ the four element group generated by the vertical and the
horizontal reflections
$\gamma_{v},\gamma_{h}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$. The polygons of
the form

$\displaystyle
P(\overline{x},\overline{y}),\;P(-\overline{x},\overline{y})=\gamma_{v}P(\overline{x},\overline{y}),\;P(\overline{x},-\overline{y})=\gamma_{h}P(\overline{x},\overline{y}),\;P(-\overline{x},-\overline{y})=\gamma_{v}\circ\gamma_{h}P(\overline{x},\overline{y})$

are called _staircase polygons_ , see Figure 1. The numbers
$x_{1},\ldots,x_{k-1}$ are called staircase lengths and $y_{2},\ldots,y_{k}$
are called staircase heights of the staircase polygon
$P(\pm\overline{x},\pm\overline{y})$. The number $x_{k}$ is called the width
and $y_{1}$ is called the height of $P(\pm\overline{x},\pm\overline{y})$.

We assume that the living space $P$ of the particle is the union of four
staircase polygons determined by the four sequences
$(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}})\in\Xi$,
$\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$ (in the symmetric case all sequences
are identical):

$P(\overline{x}^{++},\overline{y}^{++}),\;P(-\overline{x}^{-+},\overline{y}^{-+}),\;P(\overline{x}^{+-},-\overline{y}^{+-}),\;P(-\overline{x}^{--},-\overline{y}^{--}),$

such that

$y^{++}_{1}=y^{-+}_{1},\ y^{+-}_{1}=y^{--}_{1},\
x^{++}_{k(\overline{x}^{++},\overline{y}^{++})}=x^{+-}_{k(\overline{x}^{+-},\overline{y}^{+-})},\
x^{-+}_{k(\overline{x}^{-+},\overline{y}^{-+})}=x^{--}_{k(\overline{x}^{--},\overline{y}^{--})}.$

Figure 2. The living space of the particle.

The space of all such polygons we denote by $\mathscr{R}$. It is the set of
star-shaped polygons with a kernel that includes the origin and with only
vertical and horizontal boundaries. The corners
$(x^{\varsigma_{1},\varsigma_{2}}_{j},y^{\varsigma_{1},\varsigma_{2}}_{j})$
are $90^{o}$ corners and are called hereafter convex corners and
$(x^{\varsigma_{1},\varsigma_{2}}_{j},y^{\varsigma_{1},\varsigma_{2}}_{j+1})$
are $270^{o}$ and are called hereafter concave corners. The four dimensional
vector
$\\{k(\overline{x}^{\varsigma_{1},\varsigma_{2}},\overline{y}^{\varsigma_{1},\varsigma_{2}})\\}_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}$
is called the topological data of the polygon $P$, whereas the set of four
vectors,
$\\{(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}}),\varsigma_{1},\varsigma_{2}\in\\{\pm\\}\\}$
is called the numerical data of the polygon111We continue to call
$x^{\varsigma_{1},\varsigma_{2}}_{k}$/ $y^{\varsigma_{1},\varsigma_{2}}_{k}$
the length /height of a step, yet, notice that this is measuring the lengths
and heights from the axes of the corresponding axes and not of the full
polygon. $P$.

Figure 3. A projected rectangle and the staircase polygon. Cyan - the
projected rectangle. Yellow - the polygon $R^{(E,E_{1})}\cap P$ includes 3
concave corners, thus, by Theorem 1.1, $g(E,E_{1})=4$.

The level set topology of $S^{P}_{E,E_{1}}$ is determined by the properties of
$R^{(E,E_{1})}\cap P$ (see Figure 3) and can be found under mild conditions on
the potential:

######  Theorem 1.1.

Assume $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are $C^{2}$ uni-
modal potentials (satisfy condition ($\diamondsuit$ ‣ 1.2)). Let $P$ be any
polygon in $\mathscr{R}$. Then for any $E>0$, for $E_{1}\in(0,E)$, the genus
of the level set $(E_{1},E-E_{1})$, $S^{P}_{E,E_{1}}$, is given by one plus
the number of concave corners in $R^{(E,E_{1})}\cap P$ :

$\displaystyle
g(E,E_{1})=1+\sum_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}\\#\\{1\leq
k<k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}):V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k})<E_{1}<E-V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{k+1})\\}.$

Specifically, for $E>0$ the interval $E_{1}\in(0,E)$ is divided to a finite
number of segments on which the level sets have a constant genus. This
partition is non-trivial for
$E>\min_{\varsigma_{1},\varsigma_{2},k}V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k})+V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{k+1})$.
Close to the end points of $(0,E)$ the genus is $1$ whereas for $E$
sufficiently large there exists an interval of level sets, $E_{1}\in I_{max}$
with genus
$g_{max}=\sum_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})-3$
and on which for almost all $E_{1}$ values the motion is uniquely ergodic.

Figure 4. The intersection of a star-shaped polygon (grey) with four iso-
energy projected rectangles (cyan) at (a) $E=2.7$ and (b) $E=5.7$ energy
values. By Theorem 1.1, the genus of the corresponding iso-energy level sets
for the 4 shown $R^{(E,E_{1})}$ rectangles are, for increasing $E_{1}$ (a)
$g(E=2.7,E_{1})=\\{1,2,4,1\\}$ (b) $g(E=5.7,E_{1})=\\{1,2,5=g_{max},1\\}$. The
potentials here are quadratic with $\omega_{1}=1,\omega_{2}=0.8\sqrt{2}$.

Figure 5. Impact Energy-Momentum bifurcation diagram: (a) for a symmetric
cross (one concave corner with multiplicity four), b) symmetric two-steps
cross (two concave corners each with multiplicity four). Here, due to symmetry
- each blue wedge corresponds to 4 overlapping wedges, so $R^{(E,E_{1})}\cap
P$ includes $4k$ concave corners if and only if $(E,E_{1})$ is in a region
covered by $4k$ overlapping shaded blue regions. Only for these regions the
level sets genus is larger than $1$. The shaded pink (respectively light-
green) regions correspond to level sets that impact the extreme vertical
(respectively horizontal) sides. Figure 6. Impact Energy-Momentum bifurcation
diagram for the asymmetric cross shown in Figure 4 (four distinct concave
corners). The colored circles correspond to the energies of the corners of the
corresponding colored staircase polygons shown in the inset. All corners have
distinct partial energies, so $R^{(E,E_{1})}\cap P$ includes $k$ concave
corners if and only if $(E,E_{1})$ is in a region covered by $k$ overlapping
shaded blue regions. Only for these regions the level sets genus is larger
than $1$. The shaded pink (respectively light green) regions include extremal
vertical (respectively horizontal) boundaries. The two vertical lines indicate
the energies $E=2.7,5.7$ and the cyan squares on these lines correspond to the
$E_{1}$ values of the rectangles shown in Figure 4.

Figures 3 and 4 provide the geometrical interpretation of this theorem in the
configuration space, where several iso-energy rectangles are plotted on top of
an asymmetric staircase polygon $P$. Figure 4 demonstrates that the number of
concave corners that are included in $R^{(E,E_{1})}\cap P$ can vary at a fixed
energy. Figures 5 and 6 demonstrate the division of the intervals
$E_{1}\in[0,E]$ of iso-energy level sets to a finite number of intervals, each
having level sets with a fixed genus. The figures show how this division
depends on the energy $E$. In these plots, called Impact Energy-Momentum
bifurcation diagrams (IEMBD, see [15, 14]) the regions in the $(E,E_{1})$
plane at which level sets include impacts with certain parts of the boundaries
are shown. Figure 5 shows these plots for symmetric polygons and Figure 6
shows it for the asymmetric polygon of Figure 4. The grey wedge in the IEMBD
corresponds to all allowed level sets (since $E_{1}\in[0,E]$). A family of
iso-energy level sets corresponds to a vertical line in this plot. The
projected rectangles of the iso-energy level sets shown in Figure 4 at two
energies correspond to the cyan squares on the two vertical black lines of
Figure 6. Each blue colored wedge in the IEMBD corresponds to level sets that
include impacts with a concave corner of one of the polygons
$P(\overline{x}^{\varsigma_{1},\varsigma_{2}},\overline{y}^{\varsigma_{1},\varsigma_{2}})$.
In case $j$ polygons have the same concave corner we say that this wedge has
multiplicity $j$, so in Figure 5 each concave corner has multiplicity 4. The
shaded pink (respectively light-green) regions correspond to level sets that
impact the extreme vertical (respectively horizontal) boundaries a polygon.
Regions that are in the complement to the blue wedges correspond to level sets
with genus 1 (so in particular small and large ($E_{1}\approx E$) values are
included in this set). Regions that are in $k_{g}$ shaded blue wedges
(counting multiplicities) have genus $k_{g}+1$. Thus, the regions in the
intersection of all the blue wedges have the maximal genus $g_{max}$. The
intersection of this region with the two pink and two light green wedges
corresponds to level sets that, for almost all $E_{1}$, have unique ergodic
dynamics with the very mild assumption (condition ($\diamondsuit$ ‣ 1.2)) on
the potentials. As described next, with stronger assumptions on the potentials
we prove unique ergodicity for almost all level sets in the allowed region,
whereas for quadratic resonant potentials the IEMBD provides a more delicate
division to segments as will be explained in Section 1.4.

###  1.2. Deck potentials

To study the properties of the invariant measures of
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ we need additional assumptions
on the potentials. For all $z_{0}\in{\mathbb{C}}$ and $r>0$ define the ball
centered at $z_{0}$ and the droplet emanating from $z_{0}$ by:

$B(z_{0},r)=\\{z\in{\mathbb{C}}:|z-z_{0}|<r\\}\quad\text{and}\quad
C(z_{0},r)=\bigcup_{s\in(0,1]}sB(z_{0},r)\;\text{if}\;r<|z_{0}|.$

We define a special class of even potentials $V_{1},V_{2}$ denoted by $Deck$.
An even $C^{2}$-map $V:{\mathbb{R}}\to{\mathbb{R}}$ belongs to $Deck$ if

($\diamondsuit$) $\displaystyle V(0)=V^{\prime}(0)=0,\quad V^{\prime}(x)>0\
\text{for all}\ x>0\ \text{and}\ \lim_{x\to+\infty}V(x)=+\infty;$
($\heartsuit$) $\displaystyle V:(0,+\infty)\to(0,+\infty)\quad\text{is an
analytic map;}$

then $V:(0,+\infty)\to(0,+\infty)$ has a holomorphic extension
$V:U\to{\mathbb{C}}$ on an open neighborhood $U\subset{\mathbb{C}}$ of
$(0,+\infty)$;

($\clubsuit$) $\displaystyle\begin{aligned} &\text{for every $E>0$ there exist
$0<r<E$ and a bounded open set $U_{E}\subset U$}\\\ &\text{such that
$V:U_{E}\to V(U_{E})$ is biholomorphic with $C(E,r)\subset
V(U_{E})$;}\end{aligned}$ ($\spadesuit$) $\displaystyle\text{there exists}\
C_{E}>0\ \text{such that}\
\left|\frac{V^{\prime\prime}(z)V(z)}{(V^{\prime}(z))^{2}}\right|\leq C_{E}\
\text{ for all}\ z\in U_{E}.$

The class $Deck$ contains all unimodal analytic maps (i.e. satisfying
($\diamondsuit$ ‣ 1.2)), see Proposition A.1. Further examples of
$Deck$-potentials which are not analytic at $0$ (such as
$V(x)=|x|^{m}\exp(-1/|x|)$) are presented in Section A. The Deck assumption
insure that the period depends analytically on the energy. Additionally, we
will most often assume that $V\in Deck$ also satisfies

($\smiley$)
$\displaystyle\frac{V(x)V^{\prime\prime}(x)}{(V^{\prime}(x))^{2}}\geq\frac{1}{2}\
\text{for all}\ x>0,\ \text{or}$ ($\sun$) $\displaystyle V\ \text{satisfies
\eqref{eq:v} and}\ \frac{VV^{\prime\prime}}{(V^{\prime})^{2}}\neq\frac{1}{2}.$

The condition ($\smiley$ ‣ 1.2) is equivalent to $V$ being the square of a
convex function, and insures that the period is a decreasing function of the
energy, while the condition ($\sun$ ‣ 1.2) means additionally not being a
quadratic function, so the period is strictly decreasing with the energy, see
Lemma 4.4 and 4.5. For example, all non-trivial non-quadratic even polynomials
with non-negative coefficients are Deck and satisfy ($\sun$ ‣ 1.2), whereas
$V(x)=x^{2}-\sqrt{2}x^{4}+x^{6}=(x+x^{3})^{2}-(2+\sqrt{2})x^{4}$ is Deck but
does not satisfy ($\smiley$ ‣ 1.2) (cf. Proposition A.4).

###  1.3. Main results for non-quadratic Deck potentials

The main result (Theorem 1.2) says that for every energy level $E>0$ and
typical (a.e.) partial energy $E_{1}\in[0,E]$ the local flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic whenever at
least one potential is not a quadratic function or both are quadratic
functions and non-resonant (i.e. $\Omega$ is irrational222Recall that when
both potentials are quadratic the ratio of frequencies $\Omega(E,E_{1})$ does
not depend on $E$ and $E_{1}$.):

######  Theorem 1.2.

Assume $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are
$Deck$-potentials satisfying ($\smiley$ ‣ 1.2). Let $P$ be any polygon in
$\mathscr{R}$. Suppose that

  * ($\alpha$)

at least one potential $V_{1}$ or $V_{2}$ satisfies ($\sun$ ‣ 1.2) or;

  * ($\beta$)

both $V_{1},V_{2}$ are quadratic maps such that $V_{1}=\Omega^{2}V_{2}$ with
$\Omega$ irrational.

Then for every energy level $E>0$ and almost every $E_{1}\in[0,E]$ the
restricted Hamiltonian flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is
uniquely ergodic.

Notice that for small energy (i.e. satisfying (1.4)), and, in fact, for all
the grey areas in the IEMBD figures (Figures 5 and 6), the motion does not
impact the polygon walls and the above theorem trivially holds as the motion
on most of the tori is of irrational rotation (in the non-quadratic case,
$(\alpha)$, the iso-energy non-degeneracy condition holds since
$\omega_{i}^{\prime}(I_{i})\geqslant 0,i=1,2$ and at least for one oscillator
the inequality is strict). The non-trivial statement is that even when impacts
occur (the pink, light-green and blueish regions in the IEMBD figures), the
flow is usually uniquely ergodic. By the definition of unique ergodicity, the
theorem tells us that for most level sets time averages are equivalent to
phase-space average for every initial condition on these level set. The
complementary set could have periodic and quasi-periodic motion coexisting on
the same level set as in [12], [1].

###  1.4. Linear oscillators case

We study separately the case when $V_{1}$ and $V_{2}$ are quadratic. This
boundary case (in the class of $Deck$-potentials satisfying $\smiley$ ‣ 1.2)
is significantly different from the general case. In the quadratic case we
have $V_{1}=\Omega^{2}V_{2}$, and we consider rational $\Omega$ so the
harmonic motion is resonant.

If the energy level $E$ is low enough then, the impacting resonant quadratic
flow, $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ does not reach the
boundary and the motion is trivially identical to the resonant periodic linear
oscillator motion (the grey area in the IEMBD figures, before any of the blue
wedges emerge):

######  Proposition 1.3.

For energies satisfying

(1.4)
$E\leq\min\\{V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k})+V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{k+1}):\varsigma_{1},\varsigma_{2}\in\\{\pm\\},0\leq
k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\\},$

where
$x^{\varsigma_{1}\varsigma_{2}}_{0}=y^{\varsigma_{1}\varsigma_{2}}_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})+1}=0$,
the restricted Hamiltonian flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$
is identical to $(\varphi^{E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ for all
$E_{1}\in[0,E]$ and, for the impacting resonant quadratic flow it corresponds
to periodic motion.

On the other hand, a non-trivial statement, with a proof which is similar to
that of Theorem 1.2, is that if the energy level $E$ is high enough so that at
least one of the extremal horizontal or vertical boundaries is reached by
orbits in $S^{P}_{E,E_{1}}$ for all $E_{1}\in[0,E]$ then
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic for a.e.
$E_{1}$ (energies beyond which the segment $[0,E]$ is covered by the union of
the pink and light green wedges in the IEMBD figures, e.g. $E\approx 5.7$ in
Figure 6, see also the corresponding projected rectangles in Figure 4b):

######  Theorem 1.4.

If

(1.5)
$E\geq\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})+\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})$

then the impacting resonant quadratic flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic for a.e.
$E_{1}\in[0,E]$.

Studying the case of intermediate $E$ is much more complex and requires new
methods. Then the interval $[0,E]$ splits into at most countably many
intervals so that for every interval $I$ from this partition we have three
possible scenarios:

  * (ue)

the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic
for a.e. $E_{1}\in I$;

  * (cp)

the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is completely periodic
for all $E_{1}\in I$;

  * (coex)

for a.e. $E_{1}\in I$ the phase space of
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ splits into two completely
periodic cylinders and two uniquely ergodic components.

For $E>0$ let $\mathcal{J}_{E}$ denote a partition (into open intervals) of
the interval $[0,E]$ determined by the numbers

$V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k}),E-V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{k})\text{
for all }\varsigma_{1},\varsigma_{2}\in\\{\pm\\},1\leq k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}).$

In the IEMBD figures, $\mathcal{J}_{E}$ corresponds to the partition of the
vertical interval $[0,E]$ by the colored wedges. Now we formulate two results
relating to the cases (cp) and (ue). For low energies, there are intervals of
$E_{1}$ values for which no impacts occur (these are the grey regions in the
IEMBD figures):

######  Proposition 1.5.

Suppose that $I\in\mathcal{J}_{E}$ is an interval such that for every
$\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$ there exists
$1\leq{l}^{\varsigma_{1}\varsigma_{2}}\leq{k}(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})$
such that

(1.6)
$I\subset\bigcap_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}\big{[}E-V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{{l}^{\varsigma_{1}\varsigma_{2}}}),V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{{l}^{\varsigma_{1}\varsigma_{2}}})\big{]}.$

Then the impacting resonant quadratic flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is completely periodic for
every $E_{1}\in I$.

Denote by $I_{nonimp}(E)$ the collection of $E_{1}$ intervals on which no
impacts occur:

(1.7)
$I_{nonimp}(E)=\bigcup_{\begin{subarray}{c}1\leq{l}^{++}\leq{k}(\bar{x}^{++},\bar{y}^{++})\\\
1\leq{l}^{+-}\leq{k}(\bar{x}^{+-},\bar{y}^{+-})\\\
1\leq{l}^{-+}\leq{k}(\bar{x}^{-+},\bar{y}^{-+})\\\
1\leq{l}^{--}\leq{k}(\bar{x}^{--},\bar{y}^{--})\end{subarray}}\bigcap_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}\big{[}E-V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{{l}^{\varsigma_{1}\varsigma_{2}}}),V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{{l}^{\varsigma_{1}\varsigma_{2}}})\big{]}.$

For sufficiently large $E$ this set is empty, whereas for sufficiently small
$E,\ I_{nonimp}(E)=[0,E]$. For intermediate values $I_{nonimp}(E)$ may be
composed of several disjoint intervals (e.g. the grey segments for $E=3$ in
Figure 5).

When at least one of the extremal boundaries is reached (the union of the pink
and light green wedges in the IEMBD figures), similar to the general case of
Theorem 1.2, we get again unique ergodicity for a.e. $E_{1}$:

######  Theorem 1.6.

For every energy level $E>0$ and almost every

$E_{1}\in\Big{[}0,E-\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})\Big{]}\cup\Big{[}\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E\Big{]}$

the impacting resonant quadratic flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic.

Now assume that $I\in\mathcal{J}_{E}$ is an interval such that impacts occur
and not with the extremal boundaries (pure blue regions in the IEMBD figures):

(1.8) $I\subset
I_{intimp}=\big{(}E-\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}}),\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})\big{)}\backslash
I_{nonimp}(E)$

This case presents non-uniform ergodic properties and requires new
constructions. Assume that $\Omega=n/m$ with $m$, $n$ coprime natural numbers.
Let $\\{red,green\\}$ be a partition of the $(\varsigma_{1},\varsigma_{2})$
set $\\{++,+-,-+,--\\}$ into two-element set so that

(1.9) $green=(pair1,pair2)=\left\\{\begin{array}[]{cl}\\{++,+-\\}&\text{ if
$m$ is odd and $n$ is even,}\\\ \\{++,-+\\}&\text{ if $m$ is even and $n$ is
odd,}\\\ \\{++,--\\}&\text{ if $m$ and $n$ are odd.}\end{array}\right.$

Recall that a billiard in a rectangle can be reflected 3 times so that the
billiard flow is conjugated to the directed flow on the torus. By rescaling we
can always consider the flow to be in the direction $45^{o}$. Let the left
lower corner of the original rectangle denote by $(--)$, the upper left corner
by $(-+)$, the lower right corner $(+-)$ and the upper right corner by $(++)$
(see Figure 7 of Section 6.2). In the rational situation, when the impacts
from the boundary of the polygon $P$ are ignored, the torus is filled with
periodic orbits. We will see that the partition corresponds to having a
periodic orbit of the scaled torus which connects the $pair1$ and $pair2$
corners. These coloured periodic orbits induce colouring of the staircase non
extremal boundaries of
$P(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}})$,
which we call coloured sides (see Figure 2, with colouring induced by taking
$m$ odd and $n$ even as in the figures of Section 7).

Next for every $colour\in\\{green,red\\}$ let

$\displaystyle\delta^{colour}(E_{1}):=\max_{\begin{subarray}{c}\varsigma_{1}\varsigma_{2}\in
colour\\\ 1\leq
k<{k}(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\\\
V_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}})<E_{1}<E-V_{2}(y_{k+1}^{\varsigma_{1}\varsigma_{2}})\end{subarray}}\Big{(}m\arccos\sqrt{\frac{V_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}})}{E_{1}}}+n\arccos\sqrt{\frac{V_{2}(y_{k+1}^{\varsigma_{1}\varsigma_{2}})}{E-E_{1}}}\Big{)}.$

We will see that $\delta^{colour}(E_{1})$ is related to the measure of orbits
that impact the coloured sides of the polygon (see details in Section 7).
Since $\delta^{green}(E_{1})+\delta^{red}(E_{1})=\pi$ for at most countably
many $E_{1}\in I$ (see Lemma 7.2), the interval $I$ has a partition into open
intervals of two kinds $\mathcal{U}^{+}_{I}$ and $\mathcal{U}^{-}_{I}$ so that

(1.10)
$\displaystyle\begin{split}\delta^{green}(E_{1})+\delta^{red}(E_{1})>\pi&\text{
for all $E_{1}\in J$ if $J\in\mathcal{U}^{+}_{I}$, }\\\
\delta^{green}(E_{1})+\delta^{red}(E_{1})<\pi&\text{ for all $E_{1}\in J$ if
$J\in\mathcal{U}^{-}_{I}$. }\end{split}$

We will see that in the first case, there are no non-impacting orbits, whereas
in the second case there are also periodic orbits which do not impact any
coloured side. In this latter case, the impacting orbits can be divided to two
separate sets, the red/green set, consisting of orbits impacting the red/green
sides.

The following theorem asserts that in the first case, for almost all $E_{1}$,
all orbits are equi-distributed, whereas in the second case, for almost all
$E_{1}$, all orbits that impact the red/green sides are equi-distributed among
the red/green set:

######  Theorem 1.7.

Let $I\subset I_{intimp}$ of Eq. (1.8) and divide $I$ to the subintervals
$\mathcal{U}^{\pm}_{I}$ satisfying (1.10). If $J\in\mathcal{U}^{+}_{I}$ then
the impacting resonant quadratic flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic for a.e.
$E_{1}\in J$. If $J\in\mathcal{U}^{-}_{I}$ then the phase space of the flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ splits for all $E_{1}\in J$
into four components, two of which are completely periodic and for a.e.
$E_{1}\in J$ the two other components are uniquely ergodic.

This result completes the description of invariant measure of the restricted
Hamiltonian flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ for every
$E>0$ and almost every $E_{1}\in[0,E]$, when $V_{1}$, $V_{2}$ are quadratic
potentials with $V_{1}=\Omega^{2}V_{2}$ and $\Omega$ is rational. We discuss
the non-trivial implications of this theorem on ergodic averages of the
impacting resonant quadratic flow in Section 9.

###  1.5. Strategy of the proof

In Section 3, using a standard change of coordinates (as observed in [1]) we
construct an isomorphism between the restricted Hamiltonian flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ and the directional billiard
flow in direction $\pm\pi/4,\pm 3\pi/4$ on a polygon
$\mathbf{P}_{E,E_{1}}\in\mathscr{R}$. For every $E>0$ this gives a piecewise
analytic curve $E_{1}\in[0,E]\mapsto\mathbf{P}_{E,E_{1}}\in\mathscr{R}$ of
billiard flows on polygons in $\mathscr{R}$. More precisely, this curve is
analytic on every interval $I\in\mathcal{J}_{E}$. The unique ergodicity
problem for this type of curves was recently studied by the first author in
[8]. In fact, a slight modification of Theorem 4.2 in [8] (see Theorem 5.1) is
applied to curves $E\ni E_{1}\mapsto\mathbf{P}_{E,E_{1}}\in\mathscr{R}$ to
show unique ergodicity $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ for a.e.
$E_{1}\in[0,E]$ whenever at least one potential $V_{1}$ or $V_{2}$ is not a
quadratic function or both are quadratic functions with $\Omega$ irrational,
see Theorem 1.2 in Section 5. Theorem 5.1 relies on the analysis of functions
indicating the width, height, length and height of steps of staircase polygons
that make up the polygon $\mathbf{P}_{E,E_{1}}$, when the parameter $E_{1}$
varies. The relevant results involving these functions necessary for
applications of Theorem 5.1 are presented and proved in Section 4.

The case when $V_{1}$, $V_{2}$ are quadratic with $V_{1}=\Omega^{2}V_{2}$ and
$\Omega$ is rational needs a more subtle version of Theorem 5.1, this is
Theorem 7.8. Recall that any directional billiard flow on any right angle
polygon is isomorphic to the translation flow on a translation surface
obtained using so called unfolding procedure from the polygon. This leads to
the study of analytic curves of translation surfaces and their translation
flows in a fixed direction. Theorem 7.8 gives a criterion for unique
ergodicity of the translation flow for almost every translation surface lying
on such a curve (this theorem is set in an abstract framework to allow other
applications). While the idea of the proof of Theorem 7.8 is similar to
Theorem 4.2 in [8], it needs more subtle reasoning, as it involves a new type
of partition of the translation surface into polygons with sides that can be
parallel to the direction of the flow. This is the main innovation in relation
to the approach used in Theorem 5.1 (and in [8]). Another important novelty is
the use of so-called distinguished sides of partitions. The key assumption of
Theorem 7.8 is that every orbit of the directional flow (finite or half-
infinite or double-infinite) hits a distinguished side, and the key
construction in proving Theorem 1.7 is of a glued surface for which this
assumption holds.

##  2\. Oscillations in one dimension

Assume that $V_{1}(x):{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ is an even
$C^{2}$-potential satisfying ($\diamondsuit$ ‣ 1.2). Then
$x^{max}(E_{1}):[0,+\infty)\to[0,+\infty)$, the inverse of the positive branch
of $V_{1}$, is continuous and $C^{2}$ on $(0,+\infty)$ with $x^{max}(0)=0$ and
$(x^{max})^{\prime}(E_{1})>0$ for $E_{1}>0$. Similar definitions apply to
$y^{max}(E_{2})=y^{max}(E-E_{1})$.

Fix an energy level $E_{1}>0$. The particle oscillates in the interval
$[-x^{max}(E_{1}),x^{max}(E_{1})]$ wandering between the ends back and forth.
We change the space coordinate to obtain a new isomorphic model of the
oscillation in which the mass point moves periodically with
$\omega_{1}(E_{1})>0$ speed on the interval
$\psi_{1}\in[-\frac{\pi}{2},\frac{\pi}{2}]$:

(2.1)
$\frac{d\psi_{1}}{dt}=\operatorname{sgn}(p_{x})\omega_{1}(E_{1}),\>\psi(0)=0,\>\psi_{1}(\pm
x^{max}(E_{1}))=\pm\frac{\pi}{2}.$

We call these coordinates action-angle-like coordinate, as they are simply
related to the transformation to action angle coordinates, see [1]. Using the
symmetry of $V_{1}$ and the notation
$p_{x}(E_{1},x)=\pm\sqrt{2}\sqrt{E_{1}-V_{1}(x)}$ for denoting the dependence
of $p_{x}$ on position and energy, we have:

(2.2)
$\psi_{1}(x,E_{1})=\frac{2\pi}{T_{1}(E_{1})}\int_{0}^{x}\frac{1}{|p_{x}(E_{1},s)|}ds=\omega_{1}(E_{1})\int_{0}^{x}\frac{1}{\sqrt{2}\sqrt{E_{1}-V_{1}(s)}}ds,$

where $T_{1}(E_{1}),\omega_{1}(E_{1})$ are the period and frequency of the
periodic flow on the $E_{1}$ level set, i.e.:

(2.3)
$\frac{1}{4}T_{1}(E_{1})=\frac{\pi}{2\omega_{1}(E_{1})}=\int_{0}^{x^{max}(E_{1})}\frac{1}{|p_{x}(E_{1},s)|}ds=\int_{0}^{x^{max}(E_{1})}\frac{1}{\sqrt{2}\sqrt{E_{1}-V_{1}(s)}}ds.$

satisfies Eq. (2.1). Now suppose additionally that our oscillator meets an
elastic barrier at a point $x^{wall}>0$. Then its trajectories are described
by the equations (1.3) if $x\leq x^{wall}$ according to the rule that if a
trajectory meets a point $(p_{x},x^{wall})$ then it jumps to
$(-p_{x},x^{wall})$ and continues its movement in accordance with (1.3). Thus,
if $x^{wall}<x^{max}(E_{1})$ the particle oscillates in the interval
$[-x^{max}(E_{1}),x^{wall}]$ and after changing the space coordinate to the
action-angle-like coordinate, $\psi_{1}$ it oscillates with the
$\omega_{1}(E_{1})$ speed on the interval
$\psi_{1}\in[-\frac{\pi}{2},\psi_{1}(x^{wall},E_{1})]$ with elastic
reflections from the ends. The maps $\psi_{2}(y,E_{2})$ and $T_{2}(E_{2})$ are
similarly defined.

##  3\. From oscillations in dimension two to billiards on polygons

Recall that the motion in configuration space on a given level set is
restricted to the polygon $P\cap R^{(E,E_{1})}$. Using the transformation to
the $\psi$ coordinates, we find the topological and numerical data of the
corresponding polygon in the $\psi$ space.

We consider the Hamiltonian flow 1.3 restricted to the polygon
$P\in\mathscr{R}$:

$P=P(\overline{x}^{++},\overline{y}^{++})\cup
P(-\overline{x}^{-+},\overline{y}^{-+})\cup
P(\overline{x}^{+-},-\overline{y}^{+-})\cup
P(-\overline{x}^{--},-\overline{y}^{--}),$

where
$(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}})\in\Xi$
for $\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$ and we are interested in the
properties of the flow restricted to iso-energy level sets
$E_{1},E_{2}=E-E_{1}$ which we denote by
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$. Fix $E>0$ and $0<E_{1}<E$. By
the definition of $S^{P}_{E,E_{1}}$ the level set is contained in

${\mathbb{R}}^{2}\times
R^{(E,E_{1})}={\mathbb{R}}^{2}\times[-x^{max}(E_{1}),x^{max}(E_{1})]\times[-y^{max}(E-E_{1}),y^{max}(E-E_{1})].$

Let us consider new coordinates on $R^{(E,E_{1})}$ given by

$\displaystyle\psi(x,y)$
$\displaystyle=(\psi_{1}(x,E_{1}),\psi_{2}(y,E-E_{1}))$

and notice that
$\frac{d\psi(x(t),y(t))}{dt}=(\operatorname{sgn}(p_{x}(t))\omega_{1}(E_{1}),\operatorname{sgn}(p_{y}(t))\omega_{2}(E-E_{1}))$.
It follows that the flow
$\big{(}\varphi^{P,E,E_{1}}_{t}\big{)}_{t\in{\mathbb{R}}}$ in the new
coordinates coincides with the directional billiard flow on

$\mathbf{P}_{E,E_{1}}:=\psi\left(P\cap R^{(E,E_{1})}\right)$

so that the directions of its orbit are:
$(\pm\omega_{1}(E_{1}),\pm\omega_{2}(E-E_{1}))$. As $P\cap
R^{(E,E_{1})}\in\mathscr{R}$ and $\psi$ sends vertical/horizontal segments to
vertical/horizontal segments, we have $\mathbf{P}_{E,E_{1}}\in\mathscr{R}$,
namely:

$\mathbf{P}_{E,E_{1}}=\mathbf{P}^{++}_{E,E_{1}}\cup\mathbf{P}^{+-}_{E,E_{1}}\cup\mathbf{P}^{-+}_{E,E_{1}}\cup\mathbf{P}^{--}_{E,E_{1}}$

Notice that the number of corner points in each quadrant of $P\cap
R^{(E,E_{1})}$ and of $\mathbf{P}_{E,E_{1}}$ are identical, whereas their
dimensions are related by the transformation $\psi$ (which depends on $E_{1}$
and $E$). We need to find these dimensions to determine the properties of the
flow.

It is convenient to first rescale $\mathbf{P}_{E,E_{1}}$ so that the
directional motion occurs in the standard directions
$(\frac{\pi}{4},-\frac{\pi}{4},\frac{3\pi}{4},-\frac{3\pi}{4})$. Thus we scale

$\hat{\psi}_{1}(x,E_{1})=\frac{\psi_{1}(x,E_{1})}{\omega_{1}(E_{1})},\quad\hat{\psi}_{2}(x,E-E_{1})=\frac{\psi_{2}(y,E-E_{1})}{\omega_{2}(E-E_{1})},$

then $\mathbf{P}_{E,E_{1}}$ is scaled to $\mathbf{\hat{P}}_{E,E_{1}}$ and for
shorthand we omit hereafter the hats. After the rescaling,

(3.1)
$\psi_{1}(x,E_{1})=\int_{0}^{x}\frac{1}{\sqrt{2}\sqrt{E_{1}-V_{1}(s)}}ds\
\text{ for }\ E_{1}\geq V_{1}(x),$

so $\psi_{1}(x_{max}(E_{1}),E_{1})=\frac{1}{4}T_{1}(E_{1})$ and, similarly,
$\psi_{1}(x,V_{1}(x))=\frac{1}{4}T_{1}(V_{1}(x)).$

The topological and numerical data of the polygon
$\mathbf{P}_{E,E_{1}}\in\mathscr{R}$ for any $E>0$ and $E_{1}\in(0,E)$,
namely, the number of corners it has in each quadrant and their locations in
the scaled $\psi$ plane, is found by computing the sequences
$\bar{\Psi}_{i}^{\varsigma_{1},\varsigma_{2}}(E,E_{1})$ ($i=1,2$) of the
corner points of $\mathbf{P}^{\varsigma_{1},\varsigma_{2}}_{E,E_{1}}$.

By the definition of staircase polygons and condition ($\diamondsuit$ ‣ 1.2),
the sequence
$\\{V_{1}({x}_{j}^{\varsigma_{1}\varsigma_{2}})\\}_{j=1}^{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}$
is monotonically increasing and
$\\{V_{2}({y}_{j}^{\varsigma_{1}\varsigma_{2}})\\}_{j=1}^{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}$
is monotone decreasing. Hence, the number of convex corners of
$\mathbf{P}^{\varsigma_{1},\varsigma_{2}}_{E,E_{1}}$, namely the length of
$\bar{\Psi}_{i}^{\varsigma_{1},\varsigma_{2}}(E,E_{1})$, is $\\\
\max\\{1,\overline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})-\underline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})+1\\}$
where:

(3.2)
$\displaystyle\begin{split}\overline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})&=\left\\{\begin{array}[]{l}\min\\{1\leq
k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}):V_{1}({x}_{k}^{\varsigma_{1}\varsigma_{2}})\geq
E_{1}\\}\\\
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\
\text{ if }V_{1}({x}_{k}^{\varsigma_{1}\varsigma_{2}})<E_{1}\text{ for }1\leq
k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\end{array}\right.\\\
\underline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})&=\left\\{\begin{array}[]{l}\max\\{1\leq
k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}):V_{2}({y}_{k}^{\varsigma_{1}\varsigma_{2}})\geq
E-E_{1}\\}\\\ 1\text{ if
}V_{2}({y}_{k}^{\varsigma_{1}\varsigma_{2}})<E-E_{1}\text{ for }1\leq k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}).\end{array}\right.\end{split}$

If
$\underline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})\geq\overline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})$
then $\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$ is a rectangle.
Moreover,

(3.3) $\displaystyle\begin{split}&\text{if
}V_{1}({x}_{l}^{\varsigma_{1}\varsigma_{2}})\geq E_{1},\
V_{2}({y}_{l}^{\varsigma_{1}\varsigma_{2}})\geq E-E_{1}\text{ for some }1\leq
l\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\\\
&\text{ then }\underline{k}^{\varsigma_{1}\varsigma_{2}}\geq
l\geq\overline{k}^{\varsigma_{1}\varsigma_{2}}\text{ and
}\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}=P\Big{(}\frac{1}{4}\varsigma_{1}T_{1}(E_{1}),\frac{1}{4}\varsigma_{2}T_{2}(E-E_{1})\Big{)}.\end{split}$

Otherwise, we have
$\underline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})\leq\overline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})$
and

$\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}=P(\varsigma_{1}\bar{\Psi}_{1}^{\varsigma_{1}\varsigma_{2}}(E_{1}),\varsigma_{2}\bar{\Psi}_{2}^{\varsigma_{1}\varsigma_{2}}(E-E_{1})),$

where the vectors
$(\bar{\Psi}_{1}^{\varsigma_{1}\varsigma_{2}}(E_{1}),\bar{\Psi}_{2}^{\varsigma_{1}\varsigma_{2}}(E_{1}))$:

$\displaystyle{\bar{\Psi}}_{1}^{\varsigma_{1}\varsigma_{2}}(E_{1})=\\{\Psi_{1,k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\\}_{k=\underline{k}^{\varsigma_{1}\varsigma_{2}}}^{\bar{k}^{\varsigma_{1}\varsigma_{2}}},\quad\bar{\Psi}_{2}^{\varsigma_{1}\varsigma_{2}}(E-E_{1})=\\{\Psi_{2,k}^{\varsigma_{1}\varsigma_{2}}(E-E_{1})\\}_{k=\underline{k}^{\varsigma_{1}\varsigma_{2}}}^{\bar{k}^{\varsigma_{1}\varsigma_{2}}}$

are found from the sequences
$(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})$
by the $(E_{1},E)$ dependent $\psi$ transformation of the corner points that
are inside $\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$. Since, for
$k<\bar{k}^{\varsigma_{1}\varsigma_{2}}$, the horizontal motion exceeds
${x}_{k}^{\varsigma_{1}\varsigma_{2}}$ (as
$V_{1}({x}_{k}^{\varsigma_{1}\varsigma_{2}})<E_{1}$) we have

(3.4)
${\Psi}_{1,k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})=\psi_{1}({x}_{k}^{\varsigma_{1}\varsigma_{2}},E_{1}),\quad\underline{k}^{\varsigma_{1}\varsigma_{2}}\leqslant
k<\bar{k}^{\varsigma_{1}\varsigma_{2}}.$

The last value of $\bar{\Psi}_{1}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})$
depends on whether $R^{(E,E_{1})}$ intersect the extremal vertical side of
$P^{\varsigma_{1}\varsigma_{2}}$ (in the $(x,y)$ plane):

(3.5)
$\Psi_{1,\bar{k}^{\varsigma_{1}\varsigma_{2}}}(E,E_{1})=\begin{cases}\psi_{1}({x}_{\bar{k}^{\varsigma_{1}\varsigma_{2}}}^{\varsigma_{1}\varsigma_{2}},E_{1})&\text{
if }\
V_{1}({x}_{\bar{k}^{\varsigma_{1}\varsigma_{2}}}^{\varsigma_{1}\varsigma_{2}})<E_{1}\\\
\frac{1}{4}T_{1}(E_{1})&\text{ if }\
V_{1}({x}_{\bar{k}^{\varsigma_{1}\varsigma_{2}}}^{\varsigma_{1}\varsigma_{2}})\geq
E_{1}.\end{cases}$

Similarly, since for $k>\underline{k}^{\varsigma_{1}\varsigma_{2}}$, the
vertical motion exceeds ${y}_{k}^{\varsigma_{1}\varsigma_{2}}$ (since
$E-E_{1}>V_{2}({y}_{k}^{\varsigma_{1}\varsigma_{2}})$) we have

(3.6)
$\quad{\Psi}_{2,k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})=\psi_{2}({y}_{k}^{\varsigma_{1}\varsigma_{2}},E-E_{1}),\quad\underline{k}^{\varsigma_{1}\varsigma_{2}}<k\leqslant\bar{k}^{\varsigma_{1}\varsigma_{2}}.$

The first value of $\bar{\Psi}_{2}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})$
depends on whether $R^{(E,E_{1})}$ intersect the extremal horizontal side of
$P^{\varsigma_{1}\varsigma_{2}}$ (in the $(x,y)$ plane):

(3.7)
$\displaystyle\qquad\Psi_{2,\underline{k}^{\varsigma_{1}\varsigma_{2}}}(E,E_{1})$
$\displaystyle=\begin{cases}\psi_{2}({y}_{\underline{k}^{\varsigma_{1}\varsigma_{2}}}^{\varsigma_{1}\varsigma_{2}},E-E_{1})&\text{
if }\
V_{2}({y}_{\underline{k}^{\varsigma_{1}\varsigma_{2}}}^{\varsigma_{1}\varsigma_{2}})<E-E_{1}\\\
\frac{1}{4}T_{2}(E-E_{1})&\text{ if }\
V_{2}({y}_{\underline{k}^{\varsigma_{1}\varsigma_{2}}}^{\varsigma_{1}\varsigma_{2}})\geq
E-E_{1}.\end{cases}$

Summarizing, the above computations show that the topological data of the
polygon $\mathbf{P}_{E,E_{1}}$ is given by
$\\{\max\\{1,\overline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})-\underline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})+1\\}\\}_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}$
and the numerical data by
$\\{(\bar{\Psi}_{1}^{\varsigma_{1}\varsigma_{2}}(E,E_{1}),\bar{\Psi}_{2}^{\varsigma_{1}\varsigma_{2}}(E,E_{1}))\\}_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}$.

###  3.1. Regions of fixed topological data

Next, we show that the topological data of the polygons corresponding to iso-
energy level sets is fixed on a finite number of intervals of $E_{1}$ values,
and this partition depends piecewise smoothly on $E$ (so the topological data
is fixed in certain parallelograms of the IEMBD, see Figure 6).

Let $X,Y$ denote the collection of widths and heights of the steps in all
quadrants:

$\displaystyle
X:=\\{x_{k}^{\varsigma_{1}\varsigma_{2}}:\varsigma_{1},\varsigma_{2}\in\\{\pm\\},1\leq
k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\\}\subset{\mathbb{R}}_{>0}$
$\displaystyle
Y:=\\{y_{k}^{\varsigma_{1}\varsigma_{2}}:\varsigma_{1},\varsigma_{2}\in\\{\pm\\},1\leq
k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})\\}\subset{\mathbb{R}}_{>0}.$

For any $E>0$ let us consider the partition $\mathcal{J}_{E}$ (into open
intervals) of the interval $[0,E]$ determined by the numbers

$V_{1}(x),E-V_{2}(y)\text{ for all }x\in X\text{ and }y\in Y.$

Then for every $I\in\mathcal{J}_{E}$, by Eq. (3.2) the numbers
$\overline{k}^{\varsigma_{1}\varsigma_{2}}_{I}=\overline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})$
and
$\underline{k}^{\varsigma_{1}\varsigma_{2}}_{I}=\underline{k}^{\varsigma_{1}\varsigma_{2}}(E,E_{1})$
do not depend on $E_{1}\in I$. Therefore, the numerical data $E_{1}\in
I\mapsto\mathbf{P}_{E,E_{1}}\in\mathscr{R}$ is a smooth (as shown in Section
4, analytic if $V_{1},V_{2}\in Deck$) curve of polygons in $\mathscr{R}$.

######  Remark 3.1.

Fix $I=(E_{min},E_{max})\in\mathcal{J}_{E}$. Then the sets

(3.8) $X_{I}:=\\{x\in X:V_{1}(x)<E_{1}\\}\text{ and }Y_{I}:=\\{y\in
Y:V_{2}(y)<E-E_{1}\\}$

do not depend on the choice of $E_{1}\in I$.

Summarizing, in view of (3.4)-(3.7), for any $E_{1}\in I$ and
$\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$

  * •

every staircase lengths of $\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$
is of the form $\psi_{1}(x,E_{1})$ for some $x\in X_{I}$;

  * •

every staircase heights of $\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$
is of the form $\psi_{2}(y,E-E_{1})$ for some $y\in Y_{I}$;

  * •

if
$E_{min}>V_{1}(x_{{k}(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})$
then the width of $\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$ is of
the form $\psi_{1}(x,E_{1})$ for some $x\in X_{I}$, otherwise, the width of
$\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$ is of the form
$\tfrac{1}{4}T_{1}(E_{1})$;

  * •

if $E_{max}<E-V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})$ then the height of
$\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$ is of the form
$\psi_{2}(y,E-E_{1})$ for some $y\in Y_{I}$, otherwise, the height of
$\mathbf{P}^{\varsigma_{1}\varsigma_{2}}_{E,E_{1}}$ is of the form
$\tfrac{1}{4}T_{2}(E-E_{2})$.

The IEMBD figures (Figures 5 and 6) provide a graphical representation of the
above summary: the intersection of the wedges boundaries with a vertical line
provide the partition to the intervals $\mathcal{J}_{E}$, each blue wedge
corresponds to a region in which another step in the staircase (a concave
corner) is included, and the pink (respectively light green) wedges correspond
to regions in which the corresponding staircase polygons widths (respectively
heights) are of the form $\psi_{1}(x,E_{1})$ (respectively
$\psi_{2}(y,E-E_{1})$). Notice that the dependence of the above partition of
$\mathcal{J}_{E}$ is piecewise smooth in $E$: it changes exactly at the
singular $E$ values $E^{sin}=\\{E:E=V_{1}(x)+V_{2}(y),x\in X,y\in Y\\}$,
namely, at the energy values at which the wedges in the IEMBD figures emanate
and/or start to intersect each other.

##  4\. Properties of numerical data in a given topological region

In this section we present some basic properties of functions
$\psi_{1}(x,E_{1}),T_{1}(E_{1})$, defined by Eq. (3.1), (2.3) as a function of
$E_{1}$ (so $T_{1}:(0,+\infty)\to{\mathbb{R}}_{>0}$ and
$\psi_{1}(x,\cdot):[V_{1}(x),+\infty)\to{\mathbb{R}}_{>0}$). We show that when
$V_{1}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ is an even $C^{2}$-potential
satisfying ($\diamondsuit$ ‣ 1.2) the function $\psi_{1}(x,E_{1})$ is analytic
and that when $V_{1}$ is a $Deck$-potential the function $T_{1}(E_{1})$ is
analytic (the same properties apply to $\psi_{2}(y,E_{2}),T_{2}(E_{2})$ with
corresponding assumptions on $V_{2}$).

######  Proposition 4.1.

Suppose that $V_{1}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ is a
$Deck$-potential. Then the map $T_{1}:(0,+\infty)\to{\mathbb{R}}_{>0}$ given
by (2.3) is analytic and

(4.1)
$\frac{1}{4}T_{1}^{\prime}(E_{1})=\frac{1}{E_{1}\sqrt{2}}\int_{0}^{x^{max}(E_{1})}\frac{1}{\sqrt{E_{1}-V_{1}(x)}}\left(\frac{1}{2}-\frac{V_{1}^{\prime\prime}(x)V_{1}(x)}{(V_{1}^{\prime}(x))^{2}}\right)dx.$

###### Proof.

Using integration by substitution twice we have

(4.2) $\displaystyle\begin{aligned}
\frac{\sqrt{2}}{4}T_{1}(E_{1})&=\int_{0}^{x^{max}(E_{1})}\frac{1}{\sqrt{E_{1}-V_{1}(x)}}dx=|_{u=V_{1}(x),x=x^{max}(u)}\int_{0}^{E_{1}}\frac{(x^{max})^{\prime}(u)}{\sqrt{E_{1}-u}}du\\\
&=|_{s=u/E_{1}}\sqrt{E_{1}}\int_{0}^{1}\frac{(x^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}ds.\end{aligned}$

Let us consider an auxiliary map $A:(0,+\infty)\to{\mathbb{R}}_{>0}$ defined
by

(4.3)
$A(E_{1})=\sqrt{\frac{1}{8E_{1}}}T_{1}(E_{1})=\int_{0}^{1}\frac{(x^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}ds$

for $E_{1}>0$. We will show that $A$ is analytic, which obviously implies the
analyticity of $T_{1}(E_{1})$ for $E_{1}>0$. To this aim, we first establish
some properties of the function $x^{max}$ and its holomorphic extension.

Suppose that $V_{1}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ is a
$Deck$-potential. In view of ($\heartsuit$ ‣ 1.2),
$V_{1}:(0,+\infty)\to(0,+\infty)$ has a holomorphic extension
$V_{1}:U\to{\mathbb{C}}$ on an open neighborhood $U\subset{\mathbb{C}}$ of
$(0,+\infty)$ such that $V_{1}^{\prime}(z)\neq 0$ for every $z\in U$. Then
$V_{1}$ is locally invertible and its inverse functions are holomorphic (i.e.
$V_{1}$ is locally biholomorphic). More subtle assumption on the domain of
biholomorphicity is formulated in ($\clubsuit$ ‣ 1.2). By ($\clubsuit$ ‣ 1.2),
for every $E_{10}>0$ there exist $0<r<E_{10}$ and a bounded open set
$U_{E_{10}}\subset U$ such that $V_{1}:U_{E_{10}}\to V_{1}(U_{E_{10}})$ is
biholomorphic with $C(E_{10},r)\subset V_{1}(U_{E_{10}})$. Then
$z^{max}=V_{1}^{-1}:V_{1}(U_{E_{10}})\to U_{E_{10}}$ is a holomorphic
extension of $x^{max}:(0,E_{10}+r)\to{\mathbb{R}}$.

Fix any $E_{10}>0$. By the above, there exists $0<r<E_{10}$ such that the
inverse map $z^{max}$ is analytic on $E_{1}\in C(E_{10},r)$ (so $E_{1}$ is a
complex variable in the proof and $E_{10}$ is a real positive number). As

(4.4)
$\frac{(z^{max})^{\prime\prime}(E_{1})}{(z^{max})^{\prime}(E_{1})}=-\frac{V_{1}^{\prime\prime}(z^{max}(E_{1}))}{(V_{1}^{\prime}(z^{max}(E_{1})))^{2}}\;\text{
for every }\;E_{1}\in V_{1}(U_{E_{10}}),$

by the assumption ($\spadesuit$ ‣ 1.2), it follows that

(4.5)
$\left|\frac{(z^{max})^{\prime\prime}(E_{1})E_{1}}{(z^{max})^{\prime}(E_{1})}\right|\leq
C_{E_{10}}\;\text{ for every }\;E_{1}\in C(E_{10},r).$

Assume that $E_{1}\in B(E_{10},r)$ and $s\in(0,1]$. Then

$\displaystyle\left|\log\frac{|(z^{max})^{\prime}(E_{1}s)|}{|(z^{max})^{\prime}(E_{10}s)|}\right|$
$\displaystyle=\big{|}\log|(z^{max})^{\prime}(E_{1}s)|-\log{|(z^{max})^{\prime}(E_{10}s)|}\big{|}$
$\displaystyle=\left|\int_{0}^{1}\frac{d}{dt}\log{|(z^{max})^{\prime}((E_{10}+(E_{1}-E_{10})t)s)|}dt\right|$
$\displaystyle\leq\int_{0}^{1}\left|\frac{\frac{d}{dt}|(z^{max})^{\prime}((E_{10}+(E_{1}-E_{10})t)s)|}{|(z^{max})^{\prime}((E_{10}+(E_{1}-E_{10})t)s)|}\right|dt$
$\displaystyle\leq\int_{0}^{1}\frac{|(z^{max})^{\prime\prime}((E_{10}+(E_{1}-E_{10})t)s)||E_{1}-E_{10}||s|}{|(z^{max})^{\prime}((E_{10}+(E_{1}-E_{10})t)s)|}dt.$

In view of (4.5), it follows that

$\displaystyle\left|\log\frac{|(z^{max})^{\prime}(E_{1}s)|}{|(z^{max})^{\prime}(E_{10}s)|}\right|$
$\displaystyle\leq
C_{E_{10}}\int_{0}^{1}\frac{|E_{1}-E_{10}||s|}{|(E_{10}+(E_{1}-E_{10})t)s|}dt\leq
C_{E_{10}}\int_{0}^{1}\frac{|E_{1}-E_{10}|}{E_{10}-|E_{1}-E_{10}|t}dt$
$\displaystyle=C_{E_{10}}\log\frac{E_{10}}{E_{10}-|E_{1}-E_{10}|}\leq
C_{E_{10}}\log\frac{E_{10}}{E_{10}-r}.$

Hence

(4.6)
$|(z^{max})^{\prime}(E_{1}s)|\leq\left(\frac{E_{10}}{E_{10}-r}\right)^{C_{E_{10}}}(z^{max})^{\prime}(E_{10}s)$

for all $E_{1}\in B(E_{10},r)$ and $s\in(0,1]$.

We now show that the holomorphic extension $A:B(E_{10},r)\to{\mathbb{C}}$ is
given by (4.3) for complex $E_{1}$. By (4.6),

$\left|\frac{(z^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}\right|\leq
C(E_{10},r)\frac{(z^{max})^{\prime}(E_{10}s)}{\sqrt{1-s}}$

for all $E_{1}\in B(E_{10},r)$ and $s\in(0,1]$. Therefore,
$|A(E_{1})|<C(E_{10},r)|A(E_{{10}})|$, so, since $A(E_{{10}})$ is finite for
positive $E_{10}$, $A:B(E_{10},r)\to{\mathbb{C}}$ is well defined.

We now show that $A:B(E_{10},r)\to{\mathbb{C}}$ is holomorphic. For every
parameter $s\in(0,1]$ let us consider the map
$\Phi_{s}:B(E_{10},r)\to{\mathbb{C}}$ given by

$\Phi_{s}(E_{1}):=\frac{(z^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}},$

so $A(E_{1})=\int_{0}^{1}\Phi_{s}(E_{1})ds$. The map $\Phi_{s}$ is holomorphic
with

$\Phi^{\prime}_{s}(E_{1}):=\frac{(z^{max})^{\prime\prime}(E_{1}s)s}{\sqrt{1-s}}.$

In view of (4.5) and (4.6), we obtain

(4.7) $\displaystyle\begin{aligned}
|\Phi^{\prime}_{s}(E_{1})|&=\left|\frac{(z^{max})^{\prime\prime}(E_{1}s)s}{\sqrt{1-s}}\right|\leq\frac{C_{E_{10}}}{|E_{1}|}\left|\frac{(z^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}\right|\\\
&\leq\frac{C_{E_{10}}E_{10}^{C_{E_{10}}}}{(E_{10}-r)^{C_{E_{10}}+1}}\frac{(z^{max})^{\prime}(E_{10}s)}{\sqrt{1-s}}=C\Phi_{s}(E_{10})\end{aligned}$

for all $E_{1}\in B(E_{10},r)$ and $s\in(0,1]$. Since $A(E_{10})$ is finite,
$A(E_{1})=\int_{0}^{1}\Phi_{s}(E_{1})\,ds$ is holomorphic and

$A^{\prime}(E_{1})=\int_{0}^{1}\Phi^{\prime}_{s}(E_{1})\,ds=\int_{0}^{1}\frac{(z^{max})^{\prime\prime}(E_{1}s)s}{\sqrt{1-s}}\,ds.$

Using integration by substitution the same as in (4.2) (used in reverse
order), we obtain, for real $E_{1}>0$,

$\displaystyle A^{\prime}(E_{1})$
$\displaystyle=\int_{0}^{1}\frac{(x^{max})^{\prime\prime}(E_{1}s)s}{\sqrt{1-s}}\,ds=\frac{1}{E_{1}\sqrt{E_{1}}}\int_{0}^{E_{1}}\frac{(x^{max})^{\prime\prime}(u)u}{\sqrt{E_{1}-u}}\,du$
$\displaystyle=\frac{1}{E_{1}\sqrt{E_{1}}}\int_{0}^{x^{max}(E_{1})}\frac{(x^{max})^{\prime\prime}(V_{1}(x))V_{1}(x)V_{1}^{\prime}(x)}{\sqrt{E_{1}-V_{1}(x)}}\,dx.$

Therefore, by (4.4), we have

$A^{\prime}(E_{1})=-\frac{1}{E_{1}\sqrt{E_{1}}}\int_{0}^{x^{max}(E_{1})}\frac{1}{\sqrt{E_{1}-V_{1}(x)}}\frac{V_{1}^{\prime\prime}(x)V_{1}(x)}{(V_{1}^{\prime}(x))^{2}}\,dx.$

Finally, it follows that

$\displaystyle\frac{1}{4}T_{1}^{\prime}(E_{1})$
$\displaystyle=\frac{d}{dE_{1}}\Big{(}\sqrt{\frac{E_{1}}{2}}A(E_{1})\Big{)}=\frac{1}{2\sqrt{2}\sqrt{E_{1}}}A(E_{1})+\sqrt{\frac{E_{1}}{2}}A^{\prime}(E_{1})$
$\displaystyle=\frac{1}{E_{1}\sqrt{2}}\int_{0}^{x^{max}(E_{1})}\frac{1}{\sqrt{E_{1}-V_{1}(x)}}\left(\frac{1}{2}-\frac{V_{1}^{\prime\prime}(x)V_{1}(x)}{(V_{1}^{\prime}(x))^{2}}\right)dx.$

∎

######  Proposition 4.2.

Suppose that $V_{1}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ is a continuous
potential such that $V_{1}:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}_{\geq 0}$ is
strictly increasing and $V_{1}(0)=0$. Then for every $x>0$ the map
$\psi_{1}(x,\cdot):(V_{1}(x),+\infty)\to{\mathbb{R}}_{>0}$ given by (3.1) is
analytic and, for every $E_{1}>V_{1}(x)$ and $n\geq 1$

(4.8)
$\frac{d^{n}}{dE_{1}^{n}}\psi_{1}(x,E_{1})=\frac{(-1)^{n}(2n-1)!!}{2^{\frac{2n+1}{2}}}\int_{0}^{x}\frac{1}{(E_{1}-V_{1}(y))^{\frac{2n+1}{2}}}dy$

Additionally, if $V_{1}\in Deck$, for every $E_{1}>V_{1}(x)$

(4.9)
$\displaystyle\begin{split}\frac{d}{dE_{1}}\psi_{1}(x,E_{1})=&\frac{1}{E_{1}\sqrt{2}}\int_{0}^{x}\\!\frac{1}{\sqrt{E_{1}\\!-\\!V_{1}(y)}}\left(\frac{1}{2}\\!-\\!\frac{V_{1}^{\prime\prime}(y)V_{1}(y)}{(V_{1}^{\prime}(y))^{2}}\right)dy\\\
&-\frac{1}{E_{1}\sqrt{E_{1}\\!-\\!V_{1}(x)}\sqrt{2}}\frac{V_{1}(x)}{V^{\prime}_{1}(x)}\end{split}$

and

(4.10) $\lim_{E_{1}\searrow
V_{1}(x)}\frac{d}{dE_{1}}\psi_{1}(x,E_{1})=-\infty.$

###### Proof.

Take any $E_{10}>V_{1}(x)$ and let $0<r<E_{10}-V_{1}(x)$. Then for every
$y\in[0,x]$ the map $\Phi_{y}:B(E_{10},r)\to{\mathbb{C}}$ given by
$\Phi_{y}(E_{1})=\tfrac{1}{\sqrt{E_{1}-V_{1}(y)}}$ is holomorphic (taking
again $E_{1}$ to be complex) with

$\Phi^{\prime}_{y}(E_{1})=-\frac{1}{2}\frac{1}{(E_{1}-V_{1}(y))^{3/2}}\;\text{
for every }\;E_{1}\in B(E_{10},r).$

Since for all $y\in[0,x]$ and $E_{1}\in B(E_{10},r)$ we have

$|\Phi^{\prime}_{y}(E_{1})|=\frac{1}{2}\frac{1}{|E_{1}-V_{1}(y)|^{3/2}}\leq\frac{1}{2(E_{10}-V_{1}(x)-r)^{3/2}},$

it follows that $\psi_{1}(x,E_{1}):B(E_{10},r)\to{\mathbb{C}}$ is holomorphic
and

$\frac{d}{dE_{1}}\psi_{1}(x,E_{1})=\int_{0}^{x}\frac{\Phi^{\prime}_{y}(E_{1})}{\sqrt{2}}\,dy=-\frac{1}{2\sqrt{2}}\int_{0}^{x}\frac{1}{(E_{1}-V_{1}(y))^{3/2}}dy$

for every $E_{1}\in B(E_{10},r)$. This gives the analyticity of
$\psi_{1}(x,E_{1})$ and (4.8) for $n=1$. Repeating the same reasoning for
higher order derivatives we obtain

$\frac{d^{n}}{dE_{1}^{n}}\psi_{1}(x,E_{1})=\int_{0}^{x}\frac{\Phi^{(n)}_{y}(E_{1})}{\sqrt{2}}\,dy=\frac{(-1)^{n}(2n-1)!!}{2^{\frac{2n+1}{2}}}\int_{0}^{y_{0}}\frac{1}{(E_{1}-V_{1}(y))^{\frac{2n+1}{2}}}dy.$

In order to show (4.9) we first notice that using integration by substitution
as in (4.2), for every $E_{1}>V_{1}(x)$ we obtain

$\sqrt{2}\psi_{1}(x,E_{1})=\int_{0}^{x}\frac{1}{\sqrt{E_{1}-V_{1}(y)}}dy=\int_{0}^{V_{1}(x)}\frac{(x^{max})^{\prime}(u)}{\sqrt{E_{1}-u}}du=\sqrt{E_{1}}\int_{0}^{V_{1}(x)/E_{1}}\frac{(x^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}ds.$

In view of (4.7) and
$\int_{0}^{t}\frac{(x^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}ds<\int_{0}^{1}\frac{(x^{max})^{\prime}(E_{1}s)}{\sqrt{1-s}}ds=A(E_{1})<+\infty$
for every $0\leq t<1$ and $E_{1}>0$, using arguments similar to those in the
proof of Proposition 4.1, for every $E_{1}>V_{1}(x)$ we obtain

$\displaystyle\sqrt{2}\frac{d}{dE_{1}}\psi_{1}(x,E_{1})$
$\displaystyle=\frac{\psi_{1}(x,E_{1})}{E_{1}\sqrt{2}}-\sqrt{E_{1}}\frac{(x^{max})^{\prime}(V_{1}(x))}{\sqrt{1-V_{1}(x)/E_{1}}}\frac{V_{1}(x)}{E_{1}^{2}}+\sqrt{E_{1}}\int_{0}^{V_{1}(x)/E_{1}}\frac{(x^{max})^{\prime\prime}(E_{1}s)s}{\sqrt{1-s}}ds$
$\displaystyle=\frac{1}{E_{1}}\int_{0}^{x}\\!\frac{1}{\sqrt{E_{1}\\!-\\!V_{1}(y)}}\left(\frac{1}{2}\\!-\\!\frac{V_{1}^{\prime\prime}(y)V_{1}(y)}{(V_{1}^{\prime}(y))^{2}}\right)dy\\!-\\!\frac{1}{E_{1}\sqrt{E_{1}\\!-\\!V_{1}(x)}}\frac{V_{1}(x)}{V_{1}^{\prime}(x)}.$

In view of ($\spadesuit$ ‣ 1.2), for every $x>0$ we have

$\left|\frac{V_{1}^{\prime\prime}(y)V_{1}(y)}{(V^{\prime}_{1}(y))^{2}}\right|\leq
C_{x}\text{ for every }y\in(0,x].$

Therefore for every $E_{1}>V_{1}(x)$ we have

$\displaystyle\left|\int_{0}^{x}\\!\frac{1}{\sqrt{E_{1}\\!-\\!V_{1}(y)}}\left(\frac{1}{2}\\!-\\!\frac{V_{1}^{\prime\prime}(y)V_{1}(y)}{(V_{1}^{\prime}(y))^{2}}\right)dy\right|$
$\displaystyle\leq\left(\frac{1}{2}+C_{x}\right)\int_{0}^{x}\\!\frac{1}{\sqrt{V_{1}(x)\\!-\\!V_{1}(y)}}\,dy$
$\displaystyle=\frac{\sqrt{2}}{4}\left(\frac{1}{2}+C_{x}\right)T_{1}(V_{1}(x))<+\infty.$

As

$\lim_{E_{1}\searrow
V_{1}(x)}\frac{1}{E_{1}\sqrt{E_{1}-V_{1}(x)}}\frac{V_{1}(x)}{V_{1}^{\prime}(x)}=+\infty,$

this gives (4.10). ∎

Suppose that $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are even
$C^{2}$-potentials satisfying ($\diamondsuit$ ‣ 1.2). Let $E$, $x_{0}$,
$y_{0}$ be positive numbers such that $V_{1}(x_{0})+V_{2}(y_{0})<E$. In view
of Proposition 4.2,
$\psi_{1}(x_{0},\cdot):(V_{1}(x_{0}),+\infty)\to{\mathbb{R}}_{>0}$ and
$\psi_{2}(y_{0},E-\cdot):[0,E-V_{2}(y_{0}))\to{\mathbb{R}}_{>0}$ are analytic.
If additionally $V_{1},V_{2}\in Deck$ then

(4.11) $\lim_{E_{1}\searrow
V_{1}(x_{0})}\frac{d}{dE_{1}}\psi_{1}(x_{0},E_{1})=-\infty,\quad\lim_{E_{1}\nearrow
E-V_{2}(y_{0})}\frac{d}{dE_{1}}\psi_{2}(y_{0},E-E_{1})=+\infty.$

Moreover, by Proposition 4.1, $T_{1},T_{2}(E-\cdot):(0,E)\to{\mathbb{R}}_{>0}$
are also analytic.

######  Proposition 4.3.

Let $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ be two
$Deck$-potentials. Fix an energy level $E>0$. Assume that
$0<x_{1}<\ldots<x_{N}$ and $0<y_{1}<\ldots<y_{K}$ are such that
$V_{1}(x_{N})+V_{2}(y_{K})<E$. Then, for any sequence
$(\gamma_{j})_{j=0}^{N+K+1}$ with $\sum^{N+K}_{j=1}|\gamma_{j}|\neq 0$, for
all but countably many $E_{1}\in[V_{1}(x_{N}),E-V_{2}(y_{K})]$, we have:

(4.12)
$\gamma_{0}\,T_{1}(E_{1})+\sum_{j=1}^{N}\gamma_{j}\,\psi_{1}({x_{j}},E_{1})+\sum_{j=1}^{K}\gamma_{N+j}\,\psi_{2}({y_{j}},E-E_{1})+\gamma_{N+K+1}\,T_{2}(E-E_{1})\neq
0.$

###### Proof.

Suppose, contrary to our claim, that (4.12) does not hold for uncountably many
$E_{1}\in(V_{1}(x_{N}),E-V_{2}(y_{K}))$. Since $T_{1}$, $T_{2}(E-\cdot)$,
$\psi_{1}({x_{j}},\cdot)$ for $1\leq j\leq N$ and $\psi_{2}({y_{j}},E-\cdot)$,
$1\leq j\leq K$ are analytic on $(V_{1}(x_{N}),E-V_{2}(y_{K}))$, we have

(4.13)
$\gamma_{0}\,T_{1}(E_{1})+\sum_{j=1}^{N}\gamma_{j}\,\psi_{1}({x_{j}},E_{1})+\sum_{j=1}^{K}\gamma_{N+j}\,\psi_{2}({y_{j}},E-E_{1})+\gamma_{N+K+1}\,T_{2}(E-E_{1})=0$

for all $E_{1}\in(V_{1}(x_{N}),E-V_{2}(y_{K}))$. Without loss of generality we
can assume that $\gamma_{N}$ or $\gamma_{N+K}$ are non-zero. To simplify the
writing, since $T_{1}$ and $T_{2}$ are also analytic functions, we denote
$\psi_{1}({x_{0}},\cdot):=T_{1}(\cdot)$ and
$\psi_{2}({y_{K+1}},\cdot):=T_{2}(\cdot)$.

Suppose that $\gamma_{N}\neq 0$. In view of Propositions 4.1 and 4.2, the maps
$\psi_{1}({x_{j}},\cdot)$ for $0\leq j\leq N-1$ and $\psi_{2}({y_{j}},\cdot)$
for $1\leq j\leq K+1$ are analytic on $(V_{1}(x_{N-1}),E-V_{2}(y_{K}))$, in
particular at $V_{1}(x_{N})$. As $\gamma_{N}\neq 0$, if (4.13) holds, the
limit

$\displaystyle\lim_{E_{1}\searrow
V_{1}(x_{N})}\frac{d}{dE_{1}}\psi_{1}({x_{N}},E_{1})$
$\displaystyle=-\sum_{j=0}^{N-1}\frac{\gamma_{j}}{\gamma_{N}}\frac{d}{dE_{1}}\psi_{1}({x_{j}},E_{1})|_{E_{1}=V_{1}(x_{N})}$
$\displaystyle\quad-\sum_{j=1}^{K+1}\frac{\gamma_{N+j}}{\gamma_{N}}\frac{d}{dE_{1}}\psi_{2}({y_{j}},E-E_{1})|_{E_{1}=V_{1}(x_{N})}$

is finite. On the other hand, by (4.11), the limit of $\lim_{E_{1}\searrow
V_{1}(x_{N})}\frac{d}{dE_{1}}\psi_{1}({x_{N}},E_{1})$ is $-\infty$. This
contradiction completes the proof when $\gamma_{N}\neq 0$.

If $\gamma_{N+K}\neq 0$, then a contradiction follows from similar arguments
based on studying the left-side limit of
$\tfrac{d}{dE_{1}}\psi_{2}({y_{K}},E-E_{1})$ at $E-V_{2}(y_{K})$. This
completes the proof. ∎

######  Lemma 4.4.

Suppose that $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are are even
$C^{2}$-potentials satisfying ($\diamondsuit$ ‣ 1.2). Then for any
$x_{0},y_{0}>0$ we have

(4.14) $\displaystyle\frac{d}{dE_{1}}\psi_{1}({x_{0}},E_{1})<0\;\text{ for all
}\;E_{1}>V_{1}(x_{0})$ (4.15)
$\displaystyle\frac{d}{dE_{1}}\psi_{2}({x_{0}},E-E_{1})>0\;\text{ for all
}\;E_{1}\in(0,E-V_{2}(y_{0})).$

Assume additionally that $V_{1},V_{2}\in Deck$ and satisfy ($\sun$ ‣ 1.2).
Then for $i=1,2$ we have

(4.16)
$\frac{V_{i}(x)V_{i}^{\prime\prime}(x)}{(V_{i}^{\prime}(x))^{2}}>\frac{1}{2}\;\text{
for all but countably many }\;x>0$

and

(4.17) $\displaystyle\frac{d}{dE_{1}}T_{1}(E_{1})<0\;\text{ for every
}\;E_{1}>0$ (4.18) $\displaystyle\frac{d}{dE_{1}}T_{2}(E-E_{1})>0\;\text{ for
every }\;E_{1}\in(0,E).$

###### Proof.

The inequalities (4.14) and (4.15) follow immediately from (4.8). The
inequality (4.16) follows immediately from the analyticity of $V_{i}$ and
($\sun$ ‣ 1.2). Finally (4.17) and (4.18) follow immediately from (4.1) and
(4.16). ∎

######  Lemma 4.5.

Let $V_{1}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ be a $Deck$-potential. Then
$V_{1}$ satisfies ($\smiley$ ‣ 1.2) if and only if $V_{1}^{1/2}$ is convex.
The following three conditions are equivalent:

  * (a)

$V_{1}$ satisfies ($\smiley$ ‣ 1.2) and does not meet ($\sun$ ‣ 1.2);

  * (b)

$\displaystyle\frac{V_{1}(x)V_{1}^{\prime\prime}(x)}{(V_{1}^{\prime}(x))^{2}}=\frac{1}{2}$
for all $x>0$;

  * (c)

$V_{1}(x)=\frac{1}{2}\omega_{1}^{2}x^{2}$ for some $\omega_{1}>0$.

Moreover, if $V_{1}(x)=\frac{1}{2}\omega_{1}^{2}x^{2}$ then

$T_{1}(E_{1})=\frac{2\pi}{\omega_{1}}\quad\text{and}\quad\psi_{1}({x_{0}},E_{1})=\frac{1}{\omega_{1}}\arcsin\frac{\omega_{1}x_{0}}{\sqrt{2E_{1}}}=\frac{1}{\omega_{1}}\arcsin\sqrt{\frac{V_{1}(x_{0})}{E_{1}}}.$

###### Proof.

The first part of the lemma follows directly from the formula (derived by
taking derivatives of $(V_{1}^{1/m}(x))^{m}$) :

(4.19)
$\frac{V_{1}(x)V_{1}^{\prime\prime}(x)}{(V_{1}^{\prime}(x))^{2}}=\frac{m-1}{m}+\frac{V_{1}^{1/m}(x)(V_{1}^{1/m})^{\prime\prime}(x)}{m((V_{1}^{1/m})^{\prime}(x))^{2}}$

that holds for all $m\in{\mathbb{N}}$ and $x>0$. Since $V_{1}(x)$,
$V_{1}^{\prime}(x)$, $V_{1}^{1/m}(x)$ and $(V_{1}^{1/m})^{\prime}(x)$ are
positive for all $x>0$, we have

$\frac{V_{1}(x)V_{1}^{\prime\prime}(x)}{(V_{1}^{\prime}(x))^{2}}\geq\frac{m-1}{m}\;\text{
for all }\;x>0$

if and only if the map $V_{1}^{1/m}$ has non-negative second derivative on
${\mathbb{R}}\setminus\\{0\\}$.

The implications (c)$\Rightarrow$(b)$\Rightarrow$(a)$\Rightarrow$(b) are
obvious. To prove (b)$\Rightarrow$(c) suppose that
$V_{1}V_{1}^{\prime\prime}=\frac{1}{2}V_{1}^{2}$. Then

$\frac{d}{dx}\log(V_{1}^{\prime}(x))=\frac{V_{1}^{\prime\prime}(x)}{V_{1}^{\prime}(x)}=\frac{1}{2}\frac{V_{1}^{\prime}(x)}{V_{1}(x)}=\frac{d}{dx}\log\sqrt{V_{1}(x)}.$

Hence, for some $C>0$ we have $V_{1}^{\prime}(x)=C\sqrt{V_{1}(x)}$. Thus

$\frac{d}{dx}\sqrt{V_{1}(x)}=\frac{1}{2}\frac{V_{1}^{\prime}(x)}{\sqrt{V_{1}(x)}}=\frac{C}{2}.$

As $V_{1}(0)=0$, it follows that
$V_{1}(x)=(\frac{C}{2})^{2}x^{2}=\frac{1}{2}\omega_{1}^{2}x^{2}$. The form of
the maps $T_{1}(E_{1})$ and $\psi_{1}({x_{0}},E_{1})$ when
$V_{1}(x)=\frac{1}{2}\omega_{1}^{2}x^{2}$ follows from direct computations. ∎

The following lemma is the counterpart of Proposition 4.3 in the case where
all $\gamma_{j}$ for $1\leq j\leq N+K$ are zero.

######  Lemma 4.6.

Assume $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are
$Deck$-potentials satisfying ($\smiley$ ‣ 1.2) and $\gamma_{0},\gamma_{N+K+1}$
are real numbers with $|\gamma_{0}|+|\gamma_{N+K+1}|\neq 0$.

If at least one potential $V_{1}$ or $V_{2}$ satisfies ($\sun$ ‣ 1.2) (i.e. is
not quadratic) then

(4.20) $\gamma_{0}\,T_{1}(E_{1})+\gamma_{N+K+1}\,T_{2}(E-E_{1})\neq 0\text{
for all but countably many }E_{1}\in(0,E).$

If both potentials $V_{1}$ and $V_{2}$ are quadratic and

(4.21) $\gamma_{0}\,T_{1}(E_{1})+\gamma_{N+K+1}\,T_{2}(E-E_{1})=0\ \text{ for
some }\ E_{1}\in(0,E),$

then $\gamma_{0}/\gamma_{N+K+1}=-\Omega$.333If $V_{1}$ and $V_{2}$ are
quadratic then $\Omega(E,E_{1})$ does not depend on $E,E_{1}$.

###### Proof.

Suppose that $V_{1}$ is not quadratic and (4.20) does not hold. Since
$T_{1}(\cdot)$ and $T_{2}(E-\cdot)$ are analytic and take only positive
values, we have:

$\gamma_{0}\,T_{1}(E_{1})+\gamma_{N+K+1}\,T_{2}(E-E_{1})=0\text{ for all
}E_{1}\in(0,E),$

both $\gamma_{0}$ and $\gamma_{N+K+1}$ do not vanish, and
$\gamma:=-\gamma_{N+K+1}/\gamma_{0}>0$. It follows that

$\frac{d}{dE_{1}}T_{1}(E_{1})=\gamma\frac{d}{dE_{1}}T_{2}(E-E_{1})\text{ for
all }E_{1}\in(0,E).$

On the other hand, by Lemma 4.4, $\frac{d}{dE_{1}}T_{1}(E_{1})>0$ and
$\frac{d}{dE_{1}}T_{2}(E-E_{1})\leq 0$ for all $E_{1}\in E$. This gives a
contradiction.

Suppose that $V_{i}(x)=\tfrac{1}{2}\omega_{i}^{2}x^{2}$ for $i=1,2$ and (4.21)
holds, since $T_{i}=\frac{2\pi}{\omega_{i}}$, Eq. (4.21) implies
$\tfrac{\gamma_{0}}{\omega_{1}}+\tfrac{\gamma_{N+K+1}}{\omega_{2}}=0$, so

$\frac{\gamma_{0}}{\gamma_{N+K+1}}=-\frac{\omega_{1}}{\omega_{2}}=\Omega.$

∎

Recall that for $n$ real-valued $C^{n-1}$-functions $f_{1},\ldots,f_{n}$ on an
interval $I$ their Wronskian at $x\in I$ is defined by

$\mathscr{W}(f_{1},\ldots,f_{n})(x)=\det\big{[}f_{i}^{(j-1)}(x)\big{]}_{1\leq
i,j\leq n}.$

We will also deal with the bracket

$[f,g](x)=\mathscr{W}(g,f)=f^{\prime}(x)g(x)-f(x)g^{\prime}(x)$

for $C^{1}$-maps $f,g:I\to{\mathbb{R}}$.

######  Proposition 4.7.

Let $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ be even
$C^{2}$-potentials satisfying ($\diamondsuit$ ‣ 1.2). Assume that
$0<x_{1}<\ldots<x_{N}$ and $0<y_{1}<\ldots<y_{K}$ are such that
$V_{1}(x_{N})+V_{2}(y_{K})<E$. Then for all
$E_{1}\in(V_{1}(x_{N}),E-V_{2}(y_{K}))$ we have

(4.22)
$\displaystyle\mathscr{W}\big{(}\psi_{1}({x_{1}},E_{1}),\ldots,\psi_{1}({x_{N}},E_{1}),\psi_{2}({y_{1}},E-E_{1}),\ldots,\psi_{2}({y_{K}},E-E_{1})\big{)}\neq
0.$

###### Proof.

Since $\mathscr{W}$ is an alternating linear form, for all
$E_{1}\in(V_{1}(x_{N}),E-V_{2}(y_{K}))$ we have

$\displaystyle\begin{aligned}
&\mathscr{W}\big{(}\psi_{1}({x_{1}},E_{1}),\ldots,\psi_{1}({x_{N}},E_{1}),\psi_{2}({y_{1}},E\\!-\\!E_{1}),\ldots,\psi_{2}({y_{K}},E\\!-\\!E_{1})\big{)}\\\
&=\mathscr{W}\big{(}\psi_{1}({x_{1}},E_{1}),\psi_{1}({x_{2}},E_{1})-\psi_{1}({x_{1}},E_{1}),\ldots,\psi_{1}({x_{N}},E_{1})-\psi_{1}({x_{N-1}},E_{1}),\\\
&\quad\psi_{2}({y_{1}},E\\!-\\!E_{1}),\psi_{2}({y_{2}},E\\!-\\!E_{1})\\!-\\!\psi_{2}({y_{1}},E\\!-\\!E_{1}),\ldots,\psi_{2}({y_{K}},E\\!-\\!E_{1})\\!-\\!\psi_{2}({y_{K-1}},E\\!-\\!E_{1})\big{)}.\end{aligned}$

Moreover, in view of (4.8) (in Proposition 4.2), we have

$\displaystyle\frac{d^{k}}{dE_{1}^{k}}\big{(}\psi_{1}({x_{j}},E_{1})-\psi_{1}({x_{j-1}},E_{1})\big{)}=\frac{(-1)^{k}(2k-1)!!}{2^{\frac{2k+1}{2}}}\int_{x_{j-1}}^{x_{j}}\frac{1}{(E_{1}-V_{1}(s_{j}))^{\frac{2k+1}{2}}}ds_{j}$
$\displaystyle\frac{d^{k}}{dE_{1}^{k}}\big{(}\psi_{2}({y_{j}},E-E_{1})-\psi_{2}({y_{j-1}},E-E_{1})\big{)}=\frac{(2k-1)!!}{2^{\frac{2k+1}{2}}}\int_{y_{j-1}}^{y_{j}}\frac{1}{(E-E_{1}-V_{2}(u_{j}))^{\frac{2k+1}{2}}}du_{j}.$

Hence

$\displaystyle\mathscr{W}\big{(}\psi_{1}({x_{1}},E_{1}),\psi_{1}({x_{2}},E_{1})-\psi_{1}({x_{1}},E_{1}),\ldots,\psi_{1}({x_{N}},E_{1})-\psi_{1}({x_{N-1}},E_{1}),$
$\displaystyle\quad\psi_{2}({y_{1}},E\\!-\\!E_{1}),\psi_{2}({y_{2}},E\\!-\\!E_{1})\\!-\\!\psi_{2}({y_{1}},E\\!-\\!E_{1}),\ldots,\psi_{2}({y_{K}},E\\!-\\!E_{1})\\!-\\!\psi_{2}({y_{K-1}},E\\!-\\!E_{1})\big{)}$
$\displaystyle=\frac{\displaystyle\prod_{n=1}^{N+K-1}(2n-1)!!}{2^{\frac{(N+K)^{2}}{2}}}\int_{0}^{x_{1}}\ldots\int_{x_{N-1}}^{x_{N}}\int_{0}^{y_{1}}\ldots\int_{y_{K-1}}^{y_{K}}\det
C\,ds_{1}\ldots ds_{N}\,du_{1}\ldots du_{K},$

where $C=[c_{jk}]_{1\leq j,k\leq N+K}$ is an $(N+K)\times(N+K)$-matrix given
by

$c_{jk}=\left\\{\begin{array}[]{ccc}\displaystyle\frac{(-1)^{k-1}}{(E_{1}-V_{1}(s_{j}))^{\frac{2k-1}{2}}}&\text{
if }&1\leq j\leq N\\\
\displaystyle\frac{1}{(E-E_{1}-V_{2}(u_{j-N}))^{\frac{2k-1}{2}}}&\text{ if
}&N+1\leq j\leq N+K.\end{array}\right.$

Fix

$\displaystyle
s_{1}\in(0,x_{1}),\,s_{2}\in(x_{1},x_{2}),\ldots,s_{N}\in(x_{N-1},x_{N}),$
$\displaystyle
u_{1}\in(0,y_{1}),\,u_{2}\in(y_{1},y_{2}),\ldots,u_{K}\in(y_{K-1},y_{K}).$

Then

$\displaystyle V_{1}(s_{j})<V_{1}(s_{j^{\prime}})\text{ if }1\leq
j<j^{\prime}\leq N,\quad V_{2}(u_{j})<V_{2}(u_{j^{\prime}})\text{ if }1\leq
j<j^{\prime}\leq K\text{ and }$ $\displaystyle
V_{1}(s_{j})+V_{2}(u_{j^{\prime}})\leq V_{1}(x_{N})+V_{2}(y_{K})<E\text{ if
}1\leq j\leq N\text{ and }1\leq j^{\prime}\leq K.$

By the Vandermonde determinant formula, we have

$\displaystyle\det C=$
$\displaystyle\prod_{j=1}^{N}\frac{-1}{\sqrt{E_{1}-V_{1}(s_{j})}}\prod_{1\leq
j<j^{\prime}\leq
N}\Big{(}\frac{-1}{E_{1}-V_{1}(s_{j^{\prime}})}-\frac{-1}{E_{1}-V_{1}(s_{j})}\Big{)}$
$\displaystyle\cdot\prod_{j=1}^{K}\frac{1}{\sqrt{E-E_{1}-V_{2}(u_{j})}}\prod_{1\leq
j<j^{\prime}\leq
K}\Big{(}\frac{1}{E-E_{1}-V_{2}(u_{j^{\prime}})}-\frac{1}{E-E_{1}-V_{2}(u_{j})}\Big{)}$
$\displaystyle\cdot\prod_{\begin{subarray}{c}1\leq j\leq N\\\ 1\leq
j^{\prime}\leq
K\end{subarray}}\Big{(}\frac{1}{E-E_{1}-V_{2}(u_{j^{\prime}})}+\frac{1}{E_{1}-V_{1}(s_{j})}\Big{)}$
$\displaystyle=$
$\displaystyle(-1)^{\frac{N(N+1)}{2}}\prod_{j=1}^{N}\frac{1}{(E_{1}-V_{1}(s_{j}))^{N+K-\frac{1}{2}}}\prod_{j=1}^{K}\frac{1}{(E-E_{1}-V_{2}(u_{j}))^{N+K-\frac{1}{2}}}$
$\displaystyle\cdot\prod_{1\leq j<j^{\prime}\leq
N}(V_{1}(s_{j^{\prime}})-V_{1}(s_{j}))\prod_{1\leq j<j^{\prime}\leq
K}(V_{2}(u_{j^{\prime}})-V_{2}(u_{j}))$
$\displaystyle\cdot\prod_{\begin{subarray}{c}1\leq j\leq N\\\ 1\leq
j^{\prime}\leq K\end{subarray}}(E-V_{1}(s_{j})-V_{2}(u_{j^{\prime}})).$

where, by the assigned intervals of
$s_{j},s_{j^{\prime}},u_{j},u_{j^{\prime}}$, all elements under the products
signs are well defined and positive. It follows that

$(-1)^{\frac{N(N+1)}{2}}\mathscr{W}\big{(}\psi_{1}({x_{1}},E_{1}),\ldots,\psi_{1}({x_{N}},E_{1}),\psi_{2}({y_{1}},E-E_{1}),\ldots,\psi_{2}({y_{K}},E-E_{1})\big{)}>0$

for all $E_{1}\in(V_{1}(x_{N}),E-V_{2}(y_{K}))$. ∎

######  Remark 4.8.

Since all maps
$\psi_{1}({x_{1}},\cdot),\ldots,\psi_{1}({x_{N}},\cdot),\psi_{2}({y_{1}},E-\cdot),\ldots,\psi_{2}({y_{K}},E-\cdot)$
are analytic on the interval $(V_{1}(x_{N}),E-V_{2}(y_{K}))$ (see Proposition
4.2), the condition (4.22) implies that for any sequence
$(\gamma_{j})_{j=1}^{N+K}$ of at least one non-zero real numbers we have

(4.23)
$\sum_{j=1}^{N}\gamma_{j}\psi_{1}({x_{j}},E_{1})+\sum_{j=1}^{k}\gamma_{j+N}\psi_{2}({y_{j}},E-E_{1})\neq
0$

for all but countable many $E_{1}\in[V_{1}(x_{N}),E-V_{2}(y_{K})]$. Notice
that Proposition 4.3 also implies Eq. (4.23), but under the stronger Deck
conditions on the potentials.

##  5\. General criterion for unique ergodicity and its application

Now consider an interval $I$ of $E_{1}$ values on which the topological data
is fixed, so that the numerical data on $I$, as proved above, depends smoothly
on $E_{1}$. More generally let $I\ni
E_{1}\mapsto\mathbf{P}(E_{1})\in\mathscr{R}$ be a $C^{\infty}$ curve of
polygonal billiard tables in $\mathscr{R}$, i.e.

$\mathbf{P}(E_{1})=\bigcup_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}P(\overline{x}^{\varsigma_{1}\varsigma_{2}}(E_{1}),\overline{y}^{\varsigma_{1}\varsigma_{2}}(E_{1})),$

where
$x_{k}^{\varsigma_{1}\varsigma_{2}},y_{k}^{\varsigma_{1}\varsigma_{2}}:I\to{\mathbb{R}}_{>0}$
are $C^{\infty}$ maps for all $\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$,
$1\leq k\leq
k(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}})$.
Let us consider two finite sets of real $C^{\infty}$ maps on $I$ given by

$\displaystyle\mathscr{X}_{\mathbf{P}}:$
$\displaystyle=\\{x_{k}^{\varsigma_{1}\varsigma_{2}}(\cdot):\varsigma_{1},\varsigma_{2}\in\\{\pm\\},1\leq
k\leq
k(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}})\\}$
$\displaystyle\mathscr{Y}_{\mathbf{P}}:$
$\displaystyle=\\{y_{k}^{\varsigma_{1}\varsigma_{2}}(\cdot):\varsigma_{1},\varsigma_{2}\in\\{\pm\\},1\leq
k\leq
k(\overline{x}^{\varsigma_{1}\varsigma_{2}},\overline{y}^{\varsigma_{1}\varsigma_{2}})\\}.$

######  Theorem 5.1 (cf. Theorem 4.2 in [8]).

Suppose that

  * ($i$)

for any choice of integer numbers $n_{\mathbf{x}}$ for
$\mathbf{x}\in\mathscr{X}_{\mathbf{P}}$ and $m_{\mathbf{y}}$ for
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}$ such that not all of them are zero, we
have

(5.1)
$\sum_{\mathbf{x}\in\mathscr{X}_{\mathbf{P}}}n_{\mathbf{x}}\mathbf{x}(E_{1})+\sum_{\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}}m_{\mathbf{y}}\mathbf{y}(E_{1})\neq
0\text{ for a.e. }E_{1}\in I;$

  * ($ii_{+-}$)

for all $\mathbf{x}\in\mathscr{X}_{\mathbf{P}}$ and
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}$ we have
$\mathbf{x}^{\prime}(E_{1})\geq 0$ and $\mathbf{y}^{\prime}(E_{1})\leq 0$ for
all $E_{1}\in I$, moreover for at least one
$\mathbf{x}\in\mathscr{X}_{\mathbf{P}}$ or
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}$ the inequality is sharp for a.e.
$E_{1}\in I$ or;

  * ($ii_{-+}$)

for all $\mathbf{x}\in\mathscr{X}_{\mathbf{P}}$ and
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}$ we have
$\mathbf{x}^{\prime}(E_{1})\leq 0$ and $\mathbf{y}^{\prime}(E_{1})\geq 0$ for
all $E_{1}\in I$, moreover for at least one
$\mathbf{x}\in\mathscr{X}_{\mathbf{P}}$ or
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}$ the inequality is sharp for a.e.
$E_{1}\in I$.

Then for a.e. $E_{1}\in I$ the billiard flow on $\mathbf{P}(E_{1})$ in
directions $\pm\pi/4,\pm 3\pi/4$ is uniquely ergodic.

###### Proof.

We show that the above conditions imply some intermediate steps of Theorems
4.2 in [8] which are used to show that the results of Theorem 2.11 in [8]
about unique ergodicity on surfaces imply the unique ergodicity on the related
polygons.

First, we take the reference function in Theorems 4.2 to be a constant
($\ell=1$). Second, conditions ($i$) in Theorem 4.2 is used to prove the above
condition (5.1) which is then used to prove that condition ($i$) in Theorem
2.11 in [8] holds. Hence, by the same reasoning as in Theorem 4.2 in [8]
assumption ($i$) implies assumption ($i$) of Theorem 2.11 in [8].

Third, similarly, conditions ($ii$) in Theorem 4.2 are used to prove the above
conditions ($ii$) which are then used to prove that condition ($ii$) in
Theorem 2.11 in [8] holds. Hence, by the same reasoning as in Theorem 4.2 in
[8] assumption ($ii$) implies assumption ($ii$) of Theorem 2.11 in [8].

We conclude that by Theorem 2.11 in [8] the billiard flow on
$\mathbf{P}(E_{1})$ in directions $\pm\pi/4,\pm 3\pi/4$ is uniquely ergodic
for a.e. $E_{1}\in I$. ∎

Assume $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are two
$Deck$-potentials satisfying ($\smiley$ ‣ 1.2). Recall that, by Lemma 4.5, if
$V_{i}$ does not satisfy ($\sun$ ‣ 1.2), then $V_{i}$ is quadratic.

######  Theorem 5.2.

Assume $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are
$Deck$-potentials satisfying ($\smiley$ ‣ 1.2). Let $P$ be any polygon in
$\mathscr{R}$. Suppose that

  * ($\alpha$)

at least one potential $V_{1}$ or $V_{2}$ satisfies ($\sun$ ‣ 1.2) or;

  * ($\beta$)

both $V_{1},V_{2}$ are quadratic maps such that $V_{1}=\Omega^{2}V_{2}$ and
${\Omega}$ is irrational.

Then for every energy level $E>0$ and almost every $E_{1}\in[0,E]$ the
restricted Hamiltonian flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is
uniquely ergodic.

Suppose that

  * ($\gamma$)

both $V_{1},V_{2}$ are quadratic maps such that $V_{1}=\Omega^{2}V_{2}$ and
${\Omega}$ is rational.

Then for any
$E>\min\\{\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}}),\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})\\}$
and almost every

$E_{1}\in\Big{[}0,E-\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})\Big{]}\cup\Big{[}\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E\Big{]}$

the restricted Hamiltonian flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$
is uniquely ergodic.

###### Proof.

We fix an energy $E>0$. First note that we can restrict our attention to any
subinterval $I\in\mathcal{J}_{E}$. As we already have observed, for every
$E_{1}\in I$ the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is
topologically conjugated to the billiard flow in directions $\pm\pi/4,\pm
3\pi/4$ on the polygon $\mathbf{P}(E_{1}):=\mathbf{P}_{E_{1},E}=\psi(P\cap
R^{(E_{1},E)})\in\mathscr{R}$. Moreover, by Remark 3.1 and Propositions 4.1
and 4.2, we have

(5.2)
$\displaystyle\begin{split}\mathscr{X}_{\mathbf{P}}&\subset\\{\psi_{1}(x,E_{1}):{x}\in
X_{I}\\}\cup\\{\frac{1}{4}T_{1}(E_{1})\\},\\\
\mathscr{Y}_{\mathbf{P}}&\subset\\{\psi_{2}(y,E-E_{1}):{y}\in
Y_{I}\\}\cup\\{\frac{1}{4}T_{2}(E-E_{1})\\}\end{split}$

and the curve $I\ni E_{1}\mapsto\mathbf{P}(E_{1})\in\mathscr{R}$ is analytic.

Cases ($\alpha$) and ($\beta$). Assume that the sets
$\mathscr{X}_{\mathbf{P}}$, $\mathscr{Y}_{\mathbf{P}}$ do not satisfy the
condition ($i$) in Theorem 5.1. In view of Proposition 4.3, there exists a
rational positive number $\gamma>0$ such that $T_{2}(E-\cdot)=\gamma
T_{1}(\cdot)$ (since condition ($i$) involves integer coefficients). However,
by Lemma 4.6 this is impossible if ($\alpha$) is satisfied. If ($\beta$) is
satisfied then, by Lemma 4.6, we have $\gamma=-{\Omega}$ which contradicts the
irrationality of $\Omega$.

In summary, it follows that either under assumption ($\alpha$) or ($\beta$),
the condition ($i$) of Theorem 5.1 holds.

Now we verify (in both cases ($\alpha$) and ($\beta$)) the condition
($ii_{-+}$) of Theorem 5.1 holds. First suppose that ($\alpha$) holds. Then,
by Lemma 4.4, for all $\mathbf{x}\in\mathscr{X}_{\mathbf{P}}$ and
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}$ and for every $E_{1}\in I$ we have

  * •

$\mathbf{x}^{\prime}(E_{1})<0$ and $\mathbf{y}^{\prime}(E_{1})\geq 0$, if
$V_{1}$ satisfies ($\sun$ ‣ 1.2);

  * •

$\mathbf{x}^{\prime}(E_{1})\leq 0$ and $\mathbf{y}^{\prime}(E_{1})>0$, if
$V_{2}$ satisfies ($\sun$ ‣ 1.2),

so we have ($ii_{-+}$) of Theorem 5.1.

Suppose that ($\beta$) holds, then
$T_{1}^{\prime}(E_{1})=T_{2}^{\prime}(E-E_{1})=0$ for every $E_{1}\in I$.
Assume first that there is at least one impact for the level sets in $I$, so
$\mathscr{X}_{\mathbf{P}}\neq\\{\frac{1}{4}T_{1}\\}$ or
$\mathscr{Y}_{\mathbf{P}}\neq\\{\frac{1}{4}T_{2}\\}$. Then, by Lemma 4.4, for
all $\mathbf{x}\in\mathscr{X}_{\mathbf{P}}\setminus\\{\frac{1}{4}T_{1}\\}$ and
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}\setminus\\{\frac{1}{4}T_{2}\\}$ we have
$\mathbf{x}^{\prime}(E_{1})<0$ and $\mathbf{y}^{\prime}(E_{1})>0$ for every
$E_{1}\in I$, so we also have ($ii_{-+}$) of Theorem 5.1, whenever the union
of these sets is non-empty.

In summary, in both these cases the unique ergodicity of
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ for a.e. $E_{1}\in I$ follows
directly from Theorem 5.1.

Finally, when ($\beta$) holds and
$\mathscr{X}_{\mathbf{P}}=\\{\frac{1}{4}T_{1}\\}$ and
$\mathscr{Y}_{\mathbf{P}}=\\{\frac{1}{4}T_{2}\\}$ the motion on all level sets
in $I$ occurs with no impacts at all, namely, the motion corresponds to the
billiard flow on the rectangle $\mathbf{P}_{I}=\mathbf{P}(E_{1})$ in
directions $\pm\pi/4,\pm 3\pi/4$ and since $\Omega$ is irrational, the motion
is also unique ergodic for all $E_{1}\in I$.

Case ($\gamma$). We consider a subinterval $I\in\mathcal{J}_{E}$ such that
impacts occur with either all the horizontal boundaries of $P$ (the
intersection of all light green wedges in the IEMBD figures) or with all the
vertical boundaries of $P$ (the intersection of all pink regions in the IEMBD
figures):

$I\subset\Big{[}0,E-\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})\Big{]}\
\text{ or }\
I\subset\Big{[}\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E\Big{]}.$

Then, in the first case we have $Y_{I}=Y$ and in the second case $X_{I}=X$ and

(5.3) $\mathscr{Y}_{\mathbf{P}}=\\{\psi_{2}(y,E-E_{1}):{y}\in Y\\}\ \text{ or
}\ \mathscr{X}_{\mathbf{P}}\subset\\{\psi_{1}(x,E_{1}):{x}\in X\\}\ \text{
respectively}$

(see Remark 3.1). Suppose that the condition ($i$) in Theorem 5.1 does not
hold, then, in the first case we have

$k\frac{1}{4}T_{1}(E_{1})+\sum_{{x}\in
X_{I}}n_{x}\psi_{1}(x,E_{1})+\sum_{{y}\in Y_{I}}m_{y}\psi_{2}(y,E-E_{1})=0$

whereas in the second case we have

$\sum_{{x}\in X_{I}}n_{x}\psi_{1}(x,E_{1})+\sum_{{y}\in
Y_{I}}m_{y}\psi_{2}(y,E-E_{1})+k\frac{1}{4}T_{2}(E-E_{1})=0$

on a subset of positive measure, where $|k|+\sum_{x\in
X_{I}}|n_{x}|+\sum_{y\in Y_{I}}|m_{y}|\neq 0$. It follows that in both cases
at least one $n_{x}$, $x\in X_{I}$ or $m_{y}$, $y\in Y_{I}$ is non-zero. This
contradicts the conclusion of Proposition 4.3.

Finally, we check that condition ($ii_{-+}$) in Theorem 5.1 is satisfied when
($\gamma$) holds. For every $E_{1}\in I$, by Lemma 4.4, for all
$\mathbf{x}\in\mathscr{X}_{\mathbf{P}}\setminus\\{\frac{1}{4}T_{1}\\}$ and
$\mathbf{y}\in\mathscr{Y}_{\mathbf{P}}\setminus\\{\frac{1}{4}T_{2}\\}$ we have
$\mathbf{x}^{\prime}(E_{1})<0$ and $\mathbf{y}^{\prime}(E_{1})>0$. Moreover,
by (5.3), when ($\gamma$) holds, at least one of these sets is non-empty.
Hence we also have ($ii_{-+}$) in Theorem 5.1 holds. Finally the unique
ergodicity of $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ for a.e.
$E_{1}\in I$ follows directly from Theorem 5.1. This completes the proof. ∎

######  Corollary 5.3.

Suppose that $V_{1},V_{2}$ are quadratic so that $V_{1}=\Omega^{2}V_{2}$ and
${\Omega}$ is rational. If the energy is sufficiently large:

(5.4)
$E\geq\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})+\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})$

then the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely
ergodic for a.e. $E_{1}\in[0,E]$.

This is a weaker version of Theorem 1.4 and it will be helpful in the proof of
Theorem 1.4.

At the end of this section we give the following partial result, which is met
with very slight assumptions on potentials $V_{1}$ and $V_{2}$.

######  Proposition 5.4.

Suppose that $V_{1},V_{2}:{\mathbb{R}}\to{\mathbb{R}}_{\geq 0}$ are even
$C^{2}$-potentials satisfying ($\diamondsuit$ ‣ 1.2). Assume that

$E>\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})+\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}}).$

and let

$E_{1}\in
I:=\Big{[}\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E-\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})\Big{]}.$

Then the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely
ergodic for a.e. $E_{1}\in I$.

###### Proof.

The argument used at the beginning of the proof of the part ($\gamma$) in
Theorem 5.2 shows that

$\mathscr{X}_{\mathbf{P}}\subset\\{\psi_{1}(x,E_{1}):{x}\in X\\}\ \text{ and
}\ \mathscr{Y}_{\mathbf{P}}=\\{\psi_{2}(y,E-E_{1}):{y}\in Y\\}.$

Notice that Proposition 4.7 combined with Remark 4.8 shows that the condition
($i$) in Theorem 5.1 is satisfied. The negativity of the derivative for all
maps from $\mathscr{X}_{\mathbf{P}}$ and the positivity of the derivative for
all maps from $\mathscr{Y}_{\mathbf{P}}$ follow directly from Lemma 4.4. Thus
the application of Theorem 5.1 again completes the proof. ∎

The interval $I$ in Proposition 5.4 corresponds to the case of impacts with
all boundaries of $P$ (the intersection of all the coloured wedges in the
IEMBD figures), and, as will be shown in Section 6.2, this corresponds to
motion on surface of genus $g_{max}$.

##  6\. Topological data revisited

###  6.1. Short introduction to translation surfaces

Since our main criterion for unique ergodicity (Theorem 7.8) is formulated in
the framework of translation surfaces, in this section we give a short
introduction to this subject. For further background material we refer the
reader to [18], [20] and [23].

A _translation surface_ $(M,\omega)$ is a compact connected orientable
topological surface $M$, together with a finite set of points $\Sigma$ (called
_singular_ points) and an atlas of charts
$\omega=\\{\zeta_{\alpha}:U_{\alpha}\to{\mathbb{C}}:\alpha\in\mathcal{A}\\}$
on $M\setminus\Sigma$ such that every transition map
$\zeta_{\beta}\circ\zeta^{-1}_{\alpha}:\zeta_{\alpha}(U_{\alpha}\cap
U_{\beta})\to\zeta_{\beta}(U_{\alpha}\cap U_{\beta})$ is a translation, i.e.
for every connected component $C$ of $U_{\alpha}\cap U_{\beta}$ there exists
$v_{\alpha,\beta}^{C}\in{\mathbb{C}}$ such that
$\zeta_{\beta}\circ\zeta^{-1}_{\alpha}(z)=z+v^{C}_{\alpha,\beta}$ for
$z\in\zeta_{\alpha}^{-1}(C)$. All points in $M\setminus\Sigma$ are called
_regular_. For every point $x\in M$ the translation structure $\omega$ allow
us to define the total angle around $x$. If $x$ is regular then the total
angle is $2\pi$. If $\sigma$ is singular then the total angle is
$2\pi(k_{\sigma}+1)$, where $k_{\sigma}\in{\mathbb{N}}$ is the multiplicity of
$\sigma$. Then

(6.1) $\sum_{\sigma\in\Sigma}k_{\sigma}=2g-2,$

where $g$ is the genus of the surface $M$.

For every $\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}$ let $X_{\theta}$ be a
tangent vector field on $M\setminus\Sigma$ which is the pullback of the unit
constant vector field $e^{i\theta}$ on ${\mathbb{C}}$ through the charts of
the atlas. Since the derivative of any transition map is the identity, the
vector field $X_{\theta}$ is well defined on $M\setminus\Sigma$. Denote by
$(\psi^{\theta}_{t})_{t\in{\mathbb{R}}}$ the corresponding local flow, called
the translation flow on $(M,\omega)$ in direction $\theta$. The flow preserves
the measure $\lambda_{\omega}$ which is the pullback of the Lebesgue measure
on ${\mathbb{C}}$. We distinguish the vertical flow
$(\psi^{v}_{t})_{t\in{\mathbb{R}}}$, i.e. for $\theta=\pi/2$.

For every $\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}$ and a translation surface
$(M,\omega)$ denote by $(M,e^{i\theta}\omega)$ the rotated translation
surface, i.e. the new charts in $e^{i\theta}\omega$ are defined by
postcomposition of charts from $\omega$ with the rotation by $\theta$. Then
the flow $(\psi^{\theta}_{t})_{t\in{\mathbb{R}}}$ on $(M,\omega)$ coincide
with the vertical flow $(\psi^{v}_{t})_{t\in{\mathbb{R}}}$ on
$(M,e^{i(\frac{\pi}{2}-\theta)}\omega)$.

A _saddle connection_ in direction $\theta$ is an orbit segment of
$(\psi^{\theta}_{t})_{t\in{\mathbb{R}}}$ that goes from a singularity to a
singularity (possibly, the same one) and has no interior singularities. A
semi-infinite orbit of $(\psi^{\theta}_{t})_{t\in{\mathbb{R}}}$ that goes from
or to a singularity is called a _separatrix_. Recall that if $(M,\omega)$ has
no saddle connection in direction $\theta$, then the flow
$(\psi^{\theta}_{t})_{t\in{\mathbb{R}}}$ is _minimal_ , i.e. every orbit
(which can be semi-infinite or double-infinite) is dense in $M$, see [20].

###  6.2. From billiards to translation surfaces

Formally the directional billiard flow on
$\mathbf{P}(E_{1})=\mathbf{P}_{E,E_{1}}$ in directions $\pm\pi/4,\pm 3\pi/4$
acts on the union of four copies of $\mathbf{P}(E_{1})$, denoted by
$\mathbf{P}(E_{1})_{\pi/4}$, $\mathbf{P}(E_{1})_{-\pi/4}$,
$\mathbf{P}(E_{1})_{3\pi/4}$, $\mathbf{P}(E_{1})_{-3\pi/4}$. Each copy
$\mathbf{P}(E_{1})_{\vartheta}$ for $\vartheta\in\\{\pm\pi/4$, $\pm 3\pi/4\\}$
represents all unit vectors flowing in the same direction $\vartheta$. After
applying the horizontal or vertical reflection (or both) to each copy
separately, we can arrange all unit vectors to flow in the same direction
$\pi/4$. More precisely, after such transformations, all unit vectors in
$\mathbf{P}(E_{1})_{\pi/4}$, $\gamma_{h}\mathbf{P}(E_{1})_{-\pi/4}$,
$\gamma_{v}\mathbf{P}(E_{1})_{3\pi/4}$ and
$\gamma_{h}\circ\gamma_{v}\mathbf{P}(E_{1})_{-3\pi/4}$ flow in the same
direction $\pi/4$. By gluing corresponding sides of these four polygons, we
get a compact connected orientable surface $M(E_{1})$ with a translation
structure inherited from the Euclidean plan, see Figure 7. Moreover, the
directional billiard flow on $\mathbf{P}(E_{1})$ in directions $\pm\pi/4$,
$\pm 3\pi/4$ is conjugate to the translation flow
$(\psi^{\pi/4}_{t})_{t\in{\mathbb{R}}}$ in direction $\pi/4$ on the
translation surface $M(E_{1})$. This is an example of using the so called
unfolding procedure coming from [7] and [21]. Additionally, the surface
$M(E_{1})$ has a natural partition into $16$ staircase polygons
$\\{\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{\sigma_{1}\sigma_{2}}:\varsigma_{1},\varsigma_{2},\sigma_{1},\sigma_{2}\in\\{\pm\\}\\}$
so that

$\displaystyle\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{++}=\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{\pi/4},\quad\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{+-}=\gamma_{h}\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{-\pi/4},$
$\displaystyle\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{-+}=\gamma_{v}\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{3\pi/4},\quad\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{--}=\gamma_{h}\circ\gamma_{v}\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}_{-3\pi/4}.$

Figure 7. The billiard table $\mathbf{P}(E_{1})$, the translation surface
$M(E_{1})$ and the marked torus ${\mathbb{T}}_{m,n}$.

###### Proof of Theorem 1.1.

The above defined translation surface $M(E_{1})$ is formed by cutting out four
polygons, denoted by
$\mathbf{R}^{++},\mathbf{R}^{+-},\mathbf{R}^{-+},\mathbf{R}^{--}$, from the
torus and gluing their opposite sides (see white polygons in the middle part
of Figure 7).

Let us consider corners of a removed polygon
$\mathbf{R}^{\varsigma_{1}\varsigma_{2}}$. Notice that convex corners (of
$\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}$) give rise to regular points
whereas concave corners give rise to singular points of multiplicity $2$.
Indeed, each convex corner is a point on $M(E_{1})$ with the total angle
$2\pi$ (the pink point on Figure 8), whereas each concave corner is a point on
$M(E_{1})$ with the total angle $4\tfrac{3\pi}{2}=6\pi$ (the blue point on
Figure 8).

Figure 8. Singularities of $M(E_{1})$.

In view of Eq. (6.1), it follows that the genus $g(E,E_{1})$ of $M(E_{1})$
(and equivalently the genus of the level set $S^{P}_{E,E_{1}}$) is the number
of concave corners in $P\cap R^{(E,E_{1})}$ plus one:

$\displaystyle g(E,E_{1})$
$\displaystyle=1+\sum_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}\max\\{0,(\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})-\underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1}))\\}$
$\displaystyle=1+\sum_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}\\#\\{1\leq
k<k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}):V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k})<E_{1}<E-V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{k+1})\\}.$

Moreover,

$\displaystyle g(E,E_{1})$ $\displaystyle\leq
1+\sum_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}(k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})-1)$
$\displaystyle=\sum_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})-3=:g_{\max}$

and the equality holds if and only if
$\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})=k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})$
and $\underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})=1$ for any
$\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$. If

$E_{1}\in[0,\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{1}^{\varsigma_{1}\varsigma_{2}})]\cup[\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}E-V_{2}(y_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E],$

then the polygon $\mathbf{P}_{E,E_{1}}$ is a rectangle, so, for all $E$, close
to the end points of the $E_{1}$-interval, the genus of $M(E_{1})$ is one.
Suppose that

$E>\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})+\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}}).$

Then, for

$E_{1}\in
I_{\max}:=\big{[}\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E-\max_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})\big{]}\Rightarrow
g(E,E_{1})=g_{\max}.$

In view of Proposition 5.4, the flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is uniquely ergodic for a.e.
$E_{1}\in I_{\max}$. Notice that in general, the interval of $E_{1}$ values
for which $g(E,E_{1})=g_{\max}$ is larger than $I_{\max}$ as the maximal genus
interval includes cases at which all concave corners of the polygon $P$ are
inside $R^{(E,E_{1})}$ yet not all of its extremal boundaries are inside
$R^{(E,E_{1})}$. ∎

##  7\. The case when $V_{1}$ and $V_{2}$ are quadratic and $\Omega$ is
rational

In this section we deal with the remaining case when both potentials $V_{1}$
and $V_{2}$ are quadratic with $V_{1}=\Omega^{2}V_{2}$, $\Omega$ rational and
the energy $E$ is not too high, i.e. the condition (5.4) does not hold. The
rationality of $\Omega$ enables the appearance of periodic orbits and some
coexistence of periodic orbits with uniquely ergodic components.

First we focus on the case when the energy is low.

###### Proof of Proposition 1.5.

As $I$ satisfies (1.6), in view of (3.3), $R^{(E,E_{1})}\cap P$ is a rectangle
for every $E_{1}\in I$ so that its width and height are
$\tfrac{1}{2}T_{1}(E_{1})$ and $\tfrac{1}{2}T_{2}(E-E_{1})$. Since $T_{1}$ and
$T_{2}$ are constants with $T_{2}/T_{1}=\Omega$ rational, the flow
$(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ is isomorphic to the
translation flow in a rational direction on the standard torus
${\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$, so it is completely periodic for all
$E_{1}\in I$. ∎

###### Proof of Proposition 1.3.

Eq. (1.4) implies that for every $\varsigma_{1},\varsigma_{2}\in\\{\pm\\}$ we
have

$E\leq
V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k})+V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{k+1})\text{
for every }0\leq k\leq
k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}}).$

Hence, for all $E_{1}\in[0,E]$, $R^{(E,E_{1})}\cap P$ is a rectangle (with no
intersections of $P$ boundaries), so as above the motion is periodic. ∎

Propositions 1.3 and 1.5 describe completely all intervals
$I\in\mathcal{J}_{E}$ for which (cp) scenario can appear (the pure grey
regions in the IEMBD figures). We now focus on the description of all
intervals for which (ue) or (coex) scenario occur. They are presented in
Theorems 1.6 and 1.7. Unfortunately, Theorem 5.1 is not effective enough to
prove these theorems. Their proofs (presented in Section 8) need a more subtle
version of Theorem 5.1, i.e. Theorem 7.8 in Section 7.4. Theorem 7.8 helps to
prove the coexistence of periodic orbits with uniquely ergodic components for
intermediate energies. To implement this plan we must now go deeper into the
proof of Theorem 2.11 in [8] and into the framework of translation surfaces.

In the rational case there is no problem with verifying the condition
($ii_{-+}$) in Theorem 5.1. Verifying the condition ($i$) in Theorem 5.1 is
impossible. Indeed, the condition ($i$) implies the minimality of the billiard
flow (every orbit is dense) for a.e. parameter $E_{1}$. Because of the
existence of periodic orbit, the condition ($i$) cannot be met.

Suppose that $\Omega=n/m$ for some coprime natural numbers $m$ and $n$. Then
$V_{1}(x)=n^{2}c^{2}x^{2}/2$ and $V_{2}(y)=m^{2}c^{2}y^{2}/2$ for some $c>0$.
In view of Lemma 4.5, we have $T_{1}(E_{1})=\tfrac{2\pi}{n{c}}$ and
$T_{2}(E-E_{1})=\tfrac{2\pi}{m{c}}$, so

$\frac{1}{4}T_{1}(E_{1})=mC\frac{\pi}{2}\text{ and
}\frac{1}{4}T_{2}(E-E_{1})=nC\frac{\pi}{2}\text{ for }C=\frac{1}{mn{c}}.$

Moreover,

$\displaystyle\psi_{1}(x,E_{1})$
$\displaystyle=mC\arcsin\sqrt{\frac{V_{1}(x)}{E_{1}}}\text{ for }E_{1}\geq
V_{1}(x),$ $\displaystyle\psi_{2}(y,E-E_{1})$
$\displaystyle=nC\arcsin\sqrt{\frac{V_{2}(y)}{E-E_{1}}}\text{ for }E_{1}\leq
E-V_{2}(y).$

Suppose that

(7.1)
$E<\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x^{\varsigma_{1}\varsigma_{2}}_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})})+\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y^{\varsigma_{1}\varsigma_{2}}_{1}).$

For higher energies, Theorem 1.4 yields (ue)-scenario for the interval
$[0,E]$.

In fact, by Theorem 1.6, for
$E>\min\\{\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}}),\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})\\}$,
the same conclusion occurs in the intervals

$\big{[}0,E-\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}})\big{]}\text{
and
}\Big{[}\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}}),E\Big{]},$

where at least one of the extremal side of $P$ intersects $R^{(E,E_{1})}$.
Therefore, we focus our attention on $I\in\mathcal{J}_{E}$ such that

(7.2) $I\subset
I_{intimp}=\big{(}E-\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{2}(y_{1}^{\varsigma_{1}\varsigma_{2}}),\min_{\varsigma_{1},\varsigma_{2}\in\\{\pm\\}}V_{1}(x_{k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})}^{\varsigma_{1}\varsigma_{2}})\big{)}\setminus
I_{nonimp}.$

So now we consider level sets that do impact some parts of the polygon sides
but none of its extremal sides, namely, the width of the polygon
$\mathbf{P}_{E,E_{1}}$ is $\frac{1}{2}T_{1}(E_{1})$, its height is
$\frac{1}{2}T_{2}(E-E_{1})$ and $\mathbf{P}_{E,E_{1}}$ is not a rectangle.
Then the width of $M(E_{1})$ is $T_{1}(E_{1})=2\pi mC$ and its height is
$T_{2}(E-E_{1})=2\pi nC$. Let us consider the translation torus

${\mathbb{T}}_{m,n}={\mathbb{R}}^{2}/(2\pi mC{\mathbb{Z}}\times 2\pi
nC{\mathbb{Z}})$

with four marked Weierstrass points $(0,0),(\pi mC,0),(0,\pi nC),(\pi mC,\pi
nC)$ labeled by $--,+-,-+,++$ respectively, see Figure 7. Then $M(E_{1})$ can
be treated as ${\mathbb{T}}_{m,n}$ with four polygons cut out. In this $E_{1}$
interval at least one removed polygon is non-empty. Removed polygons (even
trivial) are denoted by
$\mathbf{R}^{--},\mathbf{R}^{+-},\mathbf{R}^{-+},\mathbf{R}^{++}$ as in Figure
7, i.e. $\mathbf{R}^{\varsigma_{1}\varsigma_{2}}$ is associated with the
polygon $\mathbf{P}(E_{1})^{\varsigma_{1}\varsigma_{2}}$. Each removed polygon
$\mathbf{R}^{\varsigma_{1}\varsigma_{2}}$ has only vertical and horizontal
sides, is vertically and horizontally symmetric and its center of symmetry
coincide with the marked point $\varsigma_{1}\varsigma_{2}$, see Figure 7.
Moreover, the opposite sides of each removed polygon are identified in
$M(E_{1})$.

We equip the translation torus ${\mathbb{T}}_{m,n}$ with the quotient taxicab
metric $d$ (the sum of horizontal and vertical distances). Denote by
$e^{i\pi}:{\mathbb{T}}_{m,n}\to{\mathbb{T}}_{m,n}$ the hyperelliptic
involution that fixes each marked point, namely $e^{i\pi}$ is the rotation by
$\pi$.

Let us consider periodic orbits in direction $\pi/4$ on ${\mathbb{T}}_{m,n}$
passing thorough all four marked points. There are exactly two such periodic
orbits $red$ and $green$, each of them contains two marked points (their
pairing depends on the parity of $m$ and $n$), see Figure 9. As all marked
points are labeled by $++,+-,-+,--$, this gives a partition $\\{red,green\\}$
of $\\{++,+-,-+,--\\}$ into two two-element subsets, according to this
correspondence. This is exactly the partition defined by (1.9).

For every $\underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\leq
k<\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})$ let

$D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}}):=\frac{1}{4}T_{1}(E_{1})-\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1})+\frac{1}{4}T_{2}(E-E_{1})-\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1}).$

Then
$D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})$
is the maximal distance of the four corners of
$\mathbf{R}^{\varsigma_{1},\varsigma_{2}}$ associated with
$(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})\in
P^{\varsigma_{1}\varsigma_{2}}\cap R^{(E,E_{1})}$ from the segment of the
orbit in direction $\pi/4$ passing through the marked point
$\varsigma_{1}\varsigma_{2}$ (the center of
$\mathbf{R}^{\varsigma_{1},\varsigma_{2}}$) in the time interval
$(-\min\\{m,n\\}C\pi,\min\\{m,n\\}C\pi)$. Indeed, for this segment, we can
consider the taxicab metric on ${\mathbb{R}}^{2}$, then the distance of a
point $(x_{1},y_{1})$ from the line $\\{(x_{0}+t,y_{0}+t):t\in{\mathbb{R}}\\}$
is equal to

$\min_{t}|x_{1}-x_{0}-t|+|y_{1}-y_{0}-t|=|x_{1}-x_{0}-(y_{1}-y_{0})|.$

Shifting the center of $\mathbf{R}^{\varsigma_{1}\varsigma_{2}}$ to
$(\frac{\sigma_{1}}{4}T_{1}(E_{1}),\frac{\sigma_{2}}{4}T_{2}(E-E_{1}))$ with
$\sigma_{1},\sigma_{2}\in\\{\pm\\}$, one of the corners considered is
$(\sigma_{1}\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1}),\sigma_{2}\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1}))$
and then its distance from the corresponding orbit segment is

$\Big{|}(\frac{\sigma_{1}}{4}T_{1}(E_{1})-\sigma_{1}\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1}))-(\frac{\sigma_{2}}{4}T_{2}(E-E_{1})-\sigma_{2}\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1}))\Big{|}\leq
D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})$

and equality is realized when $\sigma_{1}=-\sigma_{2}$.

By the definition of $\delta^{colour}$, for every $colour\in\\{red,green\\}$
and $E_{1}\in I$ we have

$\displaystyle
C\delta^{colour}(E_{1}):=\max_{\begin{subarray}{c}\varsigma_{1}\varsigma_{2}\in
colour\\\ \underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\leq
k<\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\end{subarray}}C\Big{(}m\arccos\sqrt{\frac{V_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}})}{E_{1}}}+n\arccos\sqrt{\frac{V_{2}(y_{k+1}^{\varsigma_{1}\varsigma_{2}})}{E-E_{1}}}\Big{)}$
$\displaystyle=\max_{\begin{subarray}{c}\varsigma_{1}\varsigma_{2}\in
colour\\\ \underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\leq
k<\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\end{subarray}}C\Big{(}m\Big{(}\frac{\pi}{2}-\arcsin\sqrt{\frac{V_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}})}{E_{1}}}\Big{)}+n\Big{(}\frac{\pi}{2}-\arcsin\sqrt{\frac{V_{2}(y_{k+1}^{\varsigma_{1}\varsigma_{2}})}{E-E_{1}}}\Big{)}\Big{)}$
$\displaystyle=\max_{\begin{subarray}{c}\varsigma_{1}\varsigma_{2}\in
colour\\\ \underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\leq
k<\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\end{subarray}}\big{(}\frac{1}{4}T_{1}(E_{1})-\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1})+\frac{1}{4}T_{2}(E-E_{1})-\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1})\big{)}$
$\displaystyle=\max_{\begin{subarray}{c}\varsigma_{1}\varsigma_{2}\in
colour\\\ \underline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\leq
k<\overline{k}^{\varsigma_{1}\varsigma_{2}}(E_{1})\end{subarray}}D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}}).$

Therefore $C\delta^{colour}(E_{1})$ measures the maximal distance of points in
removed polygons
$\mathbf{R}^{colour}:=\bigcup_{\varsigma_{1}\varsigma_{2}}\mathbf{R}^{\varsigma_{1}\varsigma_{2}}$
from the corresponding segments of the $colour$ periodic orbit. Let us
consider two closed cylinders $R$ and $G$ in ${\mathbb{T}}_{m,n}$ that consist
of points distant from the $red$ or $green$ periodic orbit by no more than
$C\delta^{red}(E_{1})$ or $C\delta^{green}(E_{1})$ respectively. Then $R$ and
$G$ are two closed cylinders in direction $\pi/4$ consisting of periodic
orbits passing through removed polygons whose centers lie on $red$ or $green$
periodic orbit respectively. Denote by $\partial R$ and $\partial G$ the
boundary of $R$ and $G$ respectively. Since all removed polygons are
$e^{i\pi}$-invariant, both cylinders $R$ and $G$ also are.

Figure 9. Two periodic orbits ($red$ and $green$) on ${\mathbb{T}}_{m,n}$ and
the corresponding cylinders on $M(E_{1})$.

######  Remark 7.1.

Note that the distance between $red$ and $green$ periodic orbits is $\pi C$.
Recall that $M(E_{1})={\mathbb{T}}_{m,n}\setminus(\mathbf{R}^{green}\cup\
\mathbf{R}^{red})$, where the opposite sides of removed polygons are
identified. Now we consider two cases:

Case 1. If $C\delta^{red}(E_{1})+C\delta^{green}(E_{1})<\pi C$, namely,
$E_{1}\in J\in\mathcal{U}_{I}^{-}$, then ${\mathbb{T}}_{m,n}\setminus(R\cup
G)$ is non-empty and consists of two open cylinders in direction $\pi/4$,
$yellow$ and $white$ on Figure 9. Since $yellow$ and $white$ cylinders do not
meet the removed polygons, they correspond to periodic cylinders $Y$ and $W$
on the translation surface $M(E_{1})$. Moreover, the flow
$(\psi_{t}^{\pi/4})_{t\in{\mathbb{R}}}$ on $M(E_{1})$ has two more invariant
sets, $R\backslash\mathbf{R}^{red}$ and $G\backslash\mathbf{R}^{green}$, which
are traces of the corresponding cylinders from ${\mathbb{T}}_{m,n}$. The sets
$R\backslash\mathbf{R}^{red},G\backslash\mathbf{R}^{green}\subset M(E_{1})$
are translation surfaces with boundary and we will analyze them later. Notice
that all $(\psi_{t}^{\pi/4})_{t\in{\mathbb{R}}}$-orbits on
$R\backslash\mathbf{R}^{red}$ and $G\backslash\mathbf{R}^{green}$ meet the
boundary of $\mathbf{R}^{red}$ and $\mathbf{R}^{green}$, and both
$\partial(R\backslash\mathbf{R}^{red})$,
$\partial(G\backslash\mathbf{R}^{green})$ consist of saddle connections for
$(\psi_{t}^{\pi/4})_{t\in{\mathbb{R}}}$ on $M(E_{1})$.

Case 2. If $C\delta^{red}(E_{1})+C\delta^{green}(E_{1})\geq\pi C$, namely,
$E_{1}\in J\in\mathcal{U}_{I}^{+}$, then $R\cup G$ fills the whole torus
${\mathbb{T}}_{m,n}$. Therefore all
$(\psi_{t}^{\pi/4})_{t\in{\mathbb{R}}}$-orbits on $M(E_{1})$ meet the boundary
of either $\mathbf{R}^{red}$ or $\mathbf{R}^{green}$ or both.

###  7.1. Periodic cylinders (case 1)

We first note some non-degeneracy properties. Recall that each removed set,
$\mathbf{R}^{colour}$, is composed of two removed polygons
$\mathbf{R}_{\varsigma_{1}\varsigma_{2}}$, $\varsigma_{1}\varsigma_{2}\in
colour$, which are associated with the
$\mathbf{P}^{\varsigma_{1}\varsigma_{2}}(E_{1}),\varsigma_{1}\varsigma_{2}\in
colour$. We first note that saddle connections between different corners of
one removed polygon $\mathbf{R}^{\varsigma_{1}\varsigma_{2}}$ rarely occur. We
then note that only when there is a degeneracy between the polygons
$P^{\varsigma_{1}\varsigma_{2}}$ it can happen that saddle connections between
the two different removed polygons persist for all $E_{1}\in I$. Finally we
note that only rarely the periodic cylinders degenerate to a line.

######  Lemma 7.2.

Let $colour\in\\{red,green\\}$.

  1. (1)

For every $\varsigma_{1}\varsigma_{2}\in colour$ and for all but countably
many $E_{1}\in I$ we have

$C\delta^{colour}(E_{1})=D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})$

for at most one $1\leq
k<k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})$.

  2. (2)

If $\varsigma_{1}\varsigma_{2}$,
$\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}$ are different elements of a
$colour$ and

(7.3) $\displaystyle\begin{aligned}
C\delta^{colour}(E_{1})&=D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})=D_{E,E_{1}}(x_{l}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}},{{y_{l+1}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}}}})\end{aligned}$

for uncountably many $E_{1}\in I$, then
$(x_{k}^{\varsigma_{1}\varsigma_{2}},y_{k+1}^{\varsigma_{1}\varsigma_{2}})=(x_{l}^{\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}},y_{l+1}^{\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}})$.

  3. (3)

Moreover,

$\delta^{red}(E_{1})+\delta^{green}(E_{1})\neq\pi$

for all but countably many $E_{1}\in(0,E)$.

In addition, all three properties are met on an open set.

###### Proof.

(1) Suppose, contrary to our claim, that there exist $k\neq l$ such that

$D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})=D_{E,E_{1}}(x_{l}^{\varsigma_{1}\varsigma_{2}},{y_{l+1}^{\varsigma_{1}\varsigma_{2}}})$

holds for uncountably many $E_{1}$. Then

$\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1})+\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1})=\psi_{1}(x_{l}^{\varsigma_{1}\varsigma_{2}},E_{1})+\psi_{2}({y_{l+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1})$

for uncountably many $E_{1}$, which contradicts Proposition 4.3.

(2) Suppose that (7.3) holds for uncountably many $E_{1}$. Then, similarly

$\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1})+\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1})=\psi_{1}(x_{l}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}},E_{1})+\psi_{2}(,{{y_{l+1}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}}}},E-E_{1})$

for uncountably many $E_{1}$. By Proposition 4.3, it follows that
$x_{k}^{\varsigma_{1}\varsigma_{2}}=x_{l}^{\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}}$
and
$y_{k+1}^{\varsigma_{1}\varsigma_{2}}=y_{l+1}^{\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}}$.

(3) Now suppose, contrary to our claim, that for uncountably many $E_{1}$

$C\delta^{red}(E_{1})+C\delta^{green}(E_{1})=D_{E,E_{1}}(x_{k}^{\varsigma_{1}\varsigma_{2}},{y_{k+1}^{\varsigma_{1}\varsigma_{2}}})+D_{E,E_{1}}(x_{l}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}},{y_{l+1}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}}})=C\pi$

for some $\varsigma_{1}\varsigma_{2}\in red$,
$\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}\in green$, $1\leq
k<k(\bar{x}^{\varsigma_{1}\varsigma_{2}},\bar{y}^{\varsigma_{1}\varsigma_{2}})$
and $1\leq
l<k(\bar{x}^{\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}},\bar{y}^{\varsigma^{\prime}_{1}\varsigma^{\prime}_{2}})$.
Then, since
$\frac{1}{4}T_{1}=mC\frac{\pi}{2},\frac{1}{4}T_{2}=nC\frac{\pi}{2}$, we have

$\displaystyle\psi_{1}(x_{k}^{\varsigma_{1}\varsigma_{2}},E_{1})+\psi_{2}({y_{k+1}^{\varsigma_{1}\varsigma_{2}}},E-E_{1})+\psi_{1}(x_{l}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}},E_{1})+\psi_{2}(y_{l+1}^{\varsigma_{1}^{\prime}\varsigma_{2}^{\prime}},E-E_{1})$
$\displaystyle=C\pi(m+n-1)=\frac{m+n-1}{2m}T_{1}(E_{1})$

holds on an uncountable subset. This again contradicts Proposition 4.3.

Finally, the complement of the set of $E_{1}$’s for which all three properties
hold is a closed set - the set of zeros for finitely many continuous maps. ∎

By Lemma 7.2, for every $E>0$ satisfying (7.1) and for every
$I\in\mathcal{J}_{E}$ satisfying (7.2), there are two families
$\mathcal{U}^{+}_{I}$ and $\mathcal{U}^{-}_{I}$ of open subintervals of $I$
such that:

  * •

for every
$E_{1}\in\bigcup_{J\in\mathcal{U}^{+}_{I}}J\cup\bigcup_{J\in\mathcal{U}^{-}_{I}}J\subset
I$ all conclusions of Lemma 7.2 hold;

  * •

$\delta^{red}(E_{1})+\delta^{green}(E_{1})>\pi$ for all $E_{1}\in J$ if
$J\in\mathcal{U}^{+}_{I}$;

  * •

$\delta^{red}(E_{1})+\delta^{green}(E_{1})<\pi$ for all $E_{1}\in J$ if
$J\in\mathcal{U}^{-}_{I}$.

Notice that by Lemma 7.2,
$\bigcup_{J\in\mathcal{U}^{+}_{I}}J\cup\bigcup_{J\in\mathcal{U}^{-}_{I}}J\subset
I$ has at most countable complement in $I$. The following result describes
fully the dynamics of the flow $(\psi^{\pi/4}_{t})_{t\in{\mathbb{R}}}$ on
$M(E_{1})$ for a.e. $E_{1}\in I$. This gives full information about dynamics
of the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ on
$S^{P}_{E,E_{1}}$ for almost every $E_{1}\in I$.

######  Theorem 7.3.

If $J\in\mathcal{U}^{+}_{I}$ then the flow
$(\psi^{\pi/4}_{t})_{t\in{\mathbb{R}}}$ on $M(E_{1})$ is uniquely ergodic for
a.e. $E_{1}\in J$. If $J\in\mathcal{U}^{-}_{I}$ then the flows
$(\psi^{\pi/4}_{t})_{t\in{\mathbb{R}}}$ on
$(R\setminus\mathbf{R}^{red})\setminus\partial(R\setminus\mathbf{R}^{red})\subset
M(E_{1})$ and on
$(G\setminus\mathbf{R}^{green})\setminus\partial(G\setminus\mathbf{R}^{green})$
are uniquely ergodic for a.e. $E_{1}\in J$ and are periodic on the complement
of these sets with motion that does not impact the polygon $P$.

Since the flow $(\psi^{\pi/4}_{t})_{t\in{\mathbb{R}}}$ on $M(E_{1})$ is
isomorphic to the flow $(\varphi^{P,E,E_{1}}_{t})_{t\in{\mathbb{R}}}$ on
$S^{P}_{E,E_{1}}$, Theorem 1.7 is a direct consequence of Theorem 7.3. To
prove this result we need a more subtle version of Theorem 2.11 in [8] applied
to $(\psi^{\pi/4}_{t})_{t\in{\mathbb{R}}}$ on some completions
$\overline{R}(E_{1})$ of
$(R\setminus\mathbf{R}^{red})\setminus\partial(R\setminus\mathbf{R}^{red})\subset
M(E_{1})$ and $\overline{G}(E_{1})$ of
$(G\setminus\mathbf{R}^{green})\setminus\partial(G\setminus\mathbf{R}^{green})$
as described below. We formulate a bit more general construction in Sections
7.2-7.4 and then return to the proof of Theorem 7.3 in Section 8.

###  7.2. Construction of completion for $J\in\mathcal{U}^{-}_{I}$

Suppose that $E_{1}\in J$ for some $J\in\mathcal{U}_{I}^{-}$, so
$\delta^{red}(E_{1})+\delta^{green}(E_{1})<\pi$. From now on we proceed only
with the set
$(R\setminus\mathbf{R}^{red})\setminus\partial(R\setminus\mathbf{R}^{red})$,
but the following construction of completion obviously works with
$(G\setminus\mathbf{R}^{green})\setminus\partial(G\setminus\mathbf{R}^{green})$.
Property (1) of Lemma 7.2 imply that for $E_{1}\in J\in\mathcal{U}^{-}_{I}$,
the boundary of the two removed polygons has one or two intersection points
with $\partial(R\setminus\mathbf{R}^{red})$ in $M(E_{1})$.

Figure 10. The subset $R\setminus\partial R$ of $M(E_{1})$ and its completion
$\overline{R}(E_{1})$.

We assume first that there is only one intersection, then we show that for two
intersections the construction of completion is almost the same. Denote by
$r\in M(E_{1})$ the single intersection point (the green point on the left
part of Figure 10). The total angle around $r$ in $R\setminus\mathbf{R}^{red}$
is $4\pi$. Recall that $R\setminus\mathbf{R}^{red}$ is a translation surface
with the boundary and its boundary consists of two saddle connection starting
and ending at the point $r$. Both saddle connection have the same direction
$\pi/4$ and the same length. In the completion $\overline{R}(E_{1})$ we
separate the beginning from the end of the saddle connections, their beginning
we denote by $r_{b}$ (the blue point on the right part of Figure 10) and their
end by $r_{e}$ (the green point on the right part of Figure 10), and then we
glue the intervals thus obtained (the dashed lines on the right part of Figure
10), creating a kind of cylinder where its lower boundary is connected to the
upper boundary through the inner green and inner blue corners. The resulting
object $\overline{R}(E_{1})$ is a compact translation surface such that the
total angle around $r_{b}$ and $r_{e}$ is $2\pi$, so they became regular
points.

Figure 11. Separation/regularization procedure

If $\partial(R\setminus\mathbf{R}^{red})$ has two intersection points,
$r_{1},r_{2}\in M(E_{1})$ with the removed polygons, by property (1) of Lemma
7.2, each one of them must belong to a distinct removed polygon. By property
(2) of Lemma 7.2, the two points are symmetric (they have the same critical
energies $(V_{1}(x_{k}),V_{2}(y_{k+1}))$). Thus, on the surface, they
correspond to corners of the two removed polygons that are at identical
horizontal and vertical distances from the removed polygon center. Hence,
$r_{2}$ is exactly at the middle point of the line connecting $r_{1}$ to
itself (namely $r_{2}=\psi^{\pi/4}_{mnC\pi}r_{1}$ and similarly
$r_{1}=\psi^{\pi/4}_{mnC\pi}r_{2}$). We carry out the separation procedure for
both intersection points $r_{1}$ and $r_{2}$. Since all four saddle connection
creating $\partial(R\setminus\mathbf{R}^{red})$ have the same length, we can
finalize gluing the pairs of relevant intervals on the boundary. In this way
we get rid of two singular points creating four regular points.

######  Remark 7.4.

Every $(\psi_{t}^{\pi/4})_{t\in{\mathbb{R}}}$ orbit on $\overline{R}(E_{1})$
hits the boundary of a removed polygon.

###  7.3. Partitions of translation surfaces into polygons

In this section we recall some basic concepts introduced in [8] and modify
them so that the completion surface, which has natural partition into polygons
with vertical and distinguished sides, can be treated with these tools.

######  Definition 1.