# Near tangent dynamics in a class of Hamiltonian impact systems

M. Pnueli, V. Rom-Kedar  M. Pnueli1, V. Rom-Kedar1,2
1 Department of Computer Science and Applied Mathematics,
The Weizmann Institute of Science, Rehovot, Israel
2 The Estrin Family Chair of Computer Science and Applied Mathematics

(August 28, 2024; August 28, 2024)

###### Abstract

Tangencies correspond to singularities of impact systems, separating between
impacting and non-impacting trajectory segments. The closure of their orbits
constitute the singularity set, which, even in the simpler billiard limit, is
known to have a complex structure. The properties of this set are studied in a
class of near integrable two degrees-of-freedom Hamiltonian impact systems.
For this class of systems, in the integrable limit, on iso-energy surfaces,
tangency appears at an isolated torus. We construct a piecewise smooth iso-
energy return map for the perturbed flow near such a tangent torus. Away from
the singularity set, this map has invariant curves, so, the singularity set is
included in a limiting singularity band. An asymptotic upper bound of this
band width is found for both non-resonant and resonant tangent tori. In the
Diophantine case the band extent to both the impacting and non-impacting
regimes is, asymptotically, of order $\epsilon$. In the resonance case its
asymptotic extent to the non-impacting regime is at most of order
$\sqrt{\epsilon}$ whereas its maximal asymptotic extent to the impacting
regime is of order $\epsilon^{2/3}$. Studying numerically a truncated standard
form of the return map which is integrable at $\epsilon=0$ and has a square-
root singularity with a singularity parameter $\alpha$, reveals that only the
strongest resonance obeys this maximal extent resonance scaling. We find that
for all but one resonant case the singularity band has connecting orbits from
its lower to its upper boundaries. The number of iterates required for an
orbit to visit close to both the lower and upper boundaries of the singularity
band diverges as $\epsilon\alpha^{2}$ is decreased, giving rise to multitude
of transient complex phenomena. On the other hand, large $\alpha$ values at a
fixed small $\epsilon$ lead to a significant reduction in this connection
time.

##  1 Background

Hamiltonian impact systems (HIS) are piecewise smooth dynamical systems. The
rules for evolution in time are the smooth Hamiltonian dynamics inside a
phase-space domain and some reflection law at the domains’ boundary [12, 8].
For example, billiards are HIS where the Hamiltonian has only a diagonal
kinetic energy term, the smooth dynamics is restricted, in the configuration
space, to lie within a table $D$, and the reflection rule is of elastic
impacts at the tables’ boundary. As for billiards, the existence of a tangent
trajectory segment in an HIS is a singular phenomenon, separating, locally,
between nearby phase-space regions - impacting and non-impacting trajectory
segments. The study of the dynamics at and near all tangent orbits is usually
difficult.

In the context of general, far from integrable, piecewise smooth dynamical
systems, the behavior near isolated tangent periodic and homoclinic orbits,
called grazing orbits, have been extensively investigated, developing a non-
smooth bifurcation theory for classes of such systems [19, 20, 21, 4, 5, 9,
10, 15, 26]. At the other extreme, grazing orbits in a one degree degrees-of-
freedom (DOF) impact oscillator system with slowly moving walls, i.e. for a
slow-fast piecewise smooth 1.5 DOF system, were shown to destroy adiabatic
invariance due to the multiple crossings between impacting and non-impacting
motion [7]. KAM stability and return maps for an harmonic oscillator subject
to a discontinuous force and time-periodic forcing were analyzed in [14, 13].

Here we study grazing orbits of a class of HIS - near integrable HIS of the
mechanical type; For mechanical Hamiltonian impact systems, the smooth motion
is governed by a smooth potential $V$ and the motion is restricted, as for
billiards, to lie within a table $D$ in the configuration space, impacting
elastically from the table’s boundary (see, e.g. [12, 11], for physical
motivation). For a general potential $V$ and a general domain $D$ such an HIS
may be neither integrable nor hyperbolic. Yet, there are certain families of
HIS of these two extreme types; Hyperbolic impact systems appear, for example,
at high energy, when the potential $V$ is uniformly close to zero and the
table $D$ corresponds to a hyperbolic billiard. Certain multi-particle systems
moving in a linear potential were proved to be hyperbolic [27, 28] as well. At
the other end, when the the billiard table respects the continuous symmetries
of the potential, one can expect Liouville integrability [24].

Once a class of Liouville integrable mechanical HIS has been identified for
certain potentials and domains, it is natural to consider small deformations
of either the potential or the table and study the properly defined near-
integrable dynamics [23, 24]. Here we study the near tangent behavior for such
near-integrable systems. In the integrable limit, the system has an isolated
tangent torus which separates between impacting and non-impacting branched
leaves of _level sets_ on an energy surface (see [24] for a more general
setup). By constructing a Poincaré section to the tangent torus which is
transverse for all trajectories of interest - impacting, tangent and non-
impacting alike - we are able to derive and analyze the piecewise smooth
return map to this section. Moreover, since the section is also transverse to
unperturbed tori that are bounded away from the tangent torus, and since under
small conservative perturbation most of these tori persist [23], a finite
neighborhood of the tangent torus remains invariant, separating between
perturbed impacting and non-impacting regimes. We call this region the
tangency band. We construct, analyse and simulate the local return map near
the tangent torus, thus providing the description of the dynamics in this
band.

The paper is ordered as follows: setup and notations are presented in section
2, along with an outline of the main results of this paper. In section 3 we
establish the properties of the singularity curve - the curve which separates
between impacting and non-impacting returns to the Poincaré section. In
section 4, a piecewise smooth return map for near tangent level sets is
constructed. In section 5 we establish bounds on the tangency band widths. In
section 6 we present basic properties and numerical simulations for a
truncated version of the return map, demonstrating the stability results of
section 5, as well as proposing various complex behaviors and long transient
structures associated with near-tangent orbits. A discussion appears in
section 7.

##  2 Set up and main results

###  2.1 Setup

Consider a smooth, separable 2 DOF mechanical Hamiltonian with a single
straight wall which is perpendicular to one of the symmetry axes:

$H=H(\cdot;\epsilon,b)=H_{int}(q_{1},p_{1},q_{2},p_{2})+\epsilon
V_{c}(q_{1},q_{2})+b\cdot{}V_{b}(q_{1};q_{1}^{w})$ (1)

The first term $H_{int}$ represents the underlying integrable structure of the
phase space:

$H_{int}=\frac{||p||^{2}}{2}+V_{int}(q_{1},q_{2})=\frac{p_{1}^{2}}{2}+V_{1}(q_{1})+\frac{p_{2}^{2}}{2}+V_{2}(q_{2})=H_{1}(q_{1},p_{1})+H_{2}(q_{2},p_{2}),$
(2)

and satisfies the S3B conditions:

######  Definition 2.1.

_Hamiltonians of the S3B (Separable, r-Smooth, Simple, Bounded level sets)
class_ are integrable, mechanical Hamiltonians of the form (2) which satisfy
the following conditions:

  1. 1.

$V_{int}=V_{1}(q_{1})+V_{2}(q_{2})$ is a sum of two potentials each depending
on one variable,

  2. 2.

$V_{1},V_{2}$ are $C^{r+1}$ smooth,

  3. 3.

$V_{1},V_{2}$ have only a finite, discrete number of simple extremum points,

  4. 4.

$V_{1},V_{2}$ are bounded from below and go to infinity as
$|q_{1}|,|q_{2}|\rightarrow{}\infty$ respectively.

By property 4, S3B Hamiltonians have bounded energy surfaces. For simplicity
of presentation, since all the analysis here is local near a regular torus of
(1), assume hereafter that each of the potentials $V_{i}(q_{i})$, $i=1,2$ has
a single, simple minimum, located at $q_{ic}=0$ such that $V_{i}(q_{ic})=0$
(and so $\min H_{int}=0$), and are convex, so in particular $q_{i}\cdot
V_{i}^{\prime}(q_{i})>0$ for $q_{i}\neq 0$ (for potentials with multiple
number of extremal points, our results apply to the behavior near each regular
torus of the smooth system belonging to a branch of the Liouville foliation of
isoenergy surfaces, see [23]).

The second term in (1), $\epsilon V_{c}$, represents a small, regular
perturbation by coupling and is assumed to be $C^{r+1}$ smooth. Thus
$V_{c}(\cdot)$ is bounded on the bounded energy surfaces (see [23]), and the
bound generally depends on $E$, the energy level. Hereafter we assume that
$E=\mathcal{O}(1)$.

The third term in (1), the singular billiard potential $V_{b}$, corresponds to
a single vertical wall, located at the coordinate $q_{1}=q_{1}^{w}$. Motion
occurs to the right of the wall, and at the wall the particle reflects
elastically, so $p_{1}\rightarrow-p_{1}$ at the wall. The elastic reflection
is represented formally as a singular energy barrier:

$V_{b}(q_{1};q_{1}^{w})=\begin{cases}0,&q_{1}>q_{1}^{w}\\\
1,&q_{1}<q_{1}^{w}\end{cases}$ (3)

where $b$ is either a fixed large number (representing an impassable wall for
all energies of interest - can also be taken as infinity) or zero
(representing the smooth Hamiltonian system without the impact). More
generally, it is possible to extend the analysis and consider a $C^{r+1}$
smooth perturbation of the wall as in [23] and a steep smooth potential
instead of a wall as in [11, 23], however, for now, we present the near
tangent analysis for the case of a single perpendicular wall and leave such
extensions to future studies. Summarizing, for $b>0$ we consider the $C^{r}$
-smooth near-integrable Hamiltonian flow in the half plane $q_{1}>q_{1}^{w}$
with elastic impacts at the vertical wall boundary:

######  Definition 2.2.

_The wall system_ is a system of the form (1), where $H_{int}$ is of the S3B
class, in Eq. (2) $V_{i}(0)=0$ and $q_{i}\cdot V_{i}^{\prime}(q_{i})>0$ for
all $q_{i}\neq 0$, $V_{1,2,c}$ are $C^{r+1}$ smooth with $r>6$, $V_{b}$ is of
the form (3), the energy is strictly below $b$ and $q_{1}^{wall}<0$.

We denote by $\Phi^{\epsilon,im}_{t}$ the wall system flow, by
$\mathcal{R}_{1}z^{w}=\mathcal{R}_{1}(q_{1}^{w},p^{w}_{1},q^{w}_{2},p^{w}_{2})=(q_{1}^{w},-p^{w}_{1},q^{w}_{2},p^{w}_{2})$
the elastic reflection from the wall (and, more generally, by
$\mathcal{R}_{i}$ the reflection of the $i$ th momentum). We also denote by
$\Phi^{\epsilon}_{t}$ the auxiliary smooth perturbed Hamiltonian flow
associated with the wall system, i.e., the smooth flow associated with (1)
when taking $b=0$ and $V_{i},i=1,2,c$ satisfy the same assumptions as in
Definition 2.2.

The integrable unperturbed smooth flow, $\Phi^{0}_{t}$ admits global action-
angle coordinates (e.g. [2]): $(J,\varphi)=S_{1}(q_{1},p_{1}),\
(I,\theta)=S_{2}(q_{2},p_{2})$, so $H_{int}(J,I)=H_{1}(J)+H_{2}(I)$, and the
corresponding dynamics on level sets $(H_{1}(J),H_{2}(I))=(E_{1},E_{2})$ are
rotations on a torus with frequencies
$\omega_{1}(J)=\frac{2\pi}{T_{1}(E_{1})},\omega_{2}(I)=\frac{2\pi}{T_{2}(E_{2})}$.
By the assumptions on $V_{i}$, the frequencies $\omega_{i}(\cdot)$ are bounded
away from zero and the inverse functions
$J=H_{1}^{-1}(E_{1}),I=H_{2}^{-1}(E_{2})$ are well defined and $C^{r}$ smooth
for all allowed level sets. Since the wall is vertical and respects the
separability symmetry of the underlying integrable Hamiltonian flow, by the
rule of elastic reflection and the symmetry of the kinetic energy term, the
partial energies $(E_{1},E_{2})$ are preserved upon impact and the motion
remains rotational on level sets. Hence, the unperturbed wall system
($\epsilon=0$ in (1)), is Liouville integrable [23, 24]. This system has
singular tangent level sets $(E_{1},E_{2})$ exactly when
$E_{1}=V_{1}(q_{1}^{w})$ (then $p_{1}=0$ at the wall), and these level sets
correspond to a torus in the allowed region of motion for all $E_{2}>0$ if and
only if $q_{1}^{w}<0$. We deal only with this case of a vertical wall in the
left half plane (the case of $q_{1}^{w}\geq 0$ is left for future works).
Then, on each energy surface with $E>V_{1}(q_{1}^{w})$ there is a unique
tangent level set $(E_{1},E_{2})=(V_{1}(q_{1}^{w}),E-V_{1}(q_{1}^{w}))$ with
the corresponding tangency action:

$I_{tan}(E)=H_{2}^{-1}(E-V_{1}(q_{1}^{w})).$ (4)

of the motion in the $(q_{2},p_{2})$ plane. Notably, for the unperturbed wall
system, the tangent torus divides the energy surface to impacting
($I<I_{tan}(E)$ ) and non-impacting ($I>I_{tan}(E)$ ) level sets.

#### The return map

The cross-section
$\Sigma=\\{(q_{1},p_{1},q_{2},p_{2})|p_{1}=0,\dot{p}_{1}<0\\}$ is transverse
to the flow, and, for any fixed $\eta$ and sufficiently small $\epsilon$, the
condition $\dot{p}_{1}<0$ is satisfied by the wall system for all
$q_{1}>\eta$. On a given energy surface, the coordinates $(q_{2},p_{2})$ on
$\Sigma_{E}=\Sigma|_{H(q,p)=E}$, are symplectic, as are the induced action-
angle coordinates $(\theta,I)=S_{2}(q_{2},p_{2})$. The unperturbed Poincaré
return map, $\mathcal{F}_{0}:\Sigma_{E}\rightarrow\Sigma_{E}$ on the half
cylinder $\Sigma_{E}$ is of the form [23]:

$\mathcal{F}_{0}:(\theta,I)\rightarrow(\bar{\theta},\bar{I}),\quad\ \text{
where}\qquad\ \begin{cases}\bar{I}=I\\\
\bar{\theta}=\theta+\Theta(I;E)\end{cases}$ (5)

and

$\Theta(I;E)=\omega_{2}(I)\cdot(T_{1}(E-H_{2}(I))-\Delta
t_{travel}(E-H_{2}(I)))=\omega_{2}(I)\cdot\tilde{T}_{1}(E-H_{2}(I))$ (6)

with

$\Delta
t_{travel}(E_{1})=\begin{cases}2\int_{q_{min}(E_{1})}^{q_{1}^{w}}\frac{dq_{1}}{\sqrt{2(E_{1}-V_{1}(q_{1}))}}&\mbox{impact
}(E_{1}>V_{1}(q_{1}^{w}))\\\ 0&\mbox{no impact, tangency,}\end{cases}$ (7)
$T_{1}(E_{1})=2\int_{q_{min}(E_{1})}^{q_{max}(E_{1})}\frac{dq_{1}}{\sqrt{2(E_{1}-V_{1}(q_{1}))}},\quad\
V_{1}(q_{min,max}(E_{1}))=E_{1}.$ (8)

Here $q_{min/max}(E_{1})$ is the minimal/maximal $q_{1}$ value of the
$H_{1}$-level set, so $q_{max}>0$ is on $\Sigma$ and, for impacting
trajectories, $q_{min}<q_{1}^{w}<0$ is outside the billiard. Notice that the
choice of the cross-section $\Sigma_{E}$ ensures that at most a single impact
with the wall may occur in between returns to the section [23]. With no loss
of generality, we set the angle coordinates so that $\phi=0$ on $\Sigma$, and
similarly, $\theta=0$ on the section $p_{2}=0,\dot{p}_{2}<0$. Let
$z_{tan}(t,\theta;E)$ denote the unperturbed tangent trajectory with energy
$E$ and phase $\theta$ at $\Sigma_{E}$, i.e.
$z_{tan}(0,\theta;E)\in\Sigma_{E},\Pi z_{tan}(0,\theta;E)=(\theta,I_{tan}(E))$
and $\Pi z$ denotes hereafter the projection of $z$ to the
$(\theta,I)=S_{2}(q_{2},p_{2})$ plane.

The perturbed return map, $\mathcal{F}_{\epsilon}$, for sufficiently small
$\epsilon$ and away from tangency, i.e. for $|I-I_{tan}(E)|>\rho$ for some
positive $\rho$, was constructed and studied in [23]. In particular, it was
established that this map is a $C^{r}$ perturbation of the integrable map
$\mathcal{F}_{0}$, and that as long as
$\rho>K\epsilon^{\alpha},\alpha<\frac{1}{2}$, and the twist condition is
satisfied, for sufficiently small $\epsilon$ it admits KAM tori, namely
invariant curves $(\theta,I^{\epsilon}(\theta))$ of the perturbed map
$\mathcal{F}_{\epsilon}$ which are bounded away from being tangent (see
Corollary 3.2 in [23]). Here, we study the behavior near the singularities of
$\mathcal{F}_{\epsilon}$, namely for trajectories which are not bounded away
from tangency.

###  2.2 Main results

The non-smoothness of $\mathcal{F}_{0}$ near the singular circle $I_{tan}(E)$
requires special care in the construction of the perturbed dynamics,
distinguishing between the return map for impacting and non-impacting
trajectory segments. The dividing line between impacting and non-impacting
segments is the singularity curve:

######  Definition 2.3.

The _singularity curve_ ,
$\sigma^{\epsilon}=\sigma^{\epsilon}(E)=\\{(\theta^{\epsilon}_{tan}(s),I^{\epsilon}_{tan}(s))\\}_{s\in
S}$, is the set of all initial conditions in $\Sigma_{E}$ which, under the
wall system flow, touch the wall tangentially before their first return to
$\Sigma_{E}$.

######  Definition 2.4.

_The singularity set_ ,
$\mathcal{S}^{\epsilon}=\bigcup_{k=-\infty}^{\infty}\mathcal{F}_{\epsilon}^{k}\sigma^{\epsilon}$,
is the invariant set of all backward and forward iterations of the singularity
curve. Its closure is denoted by $\mathcal{\bar{S}}^{\epsilon}$.

The parametric representation for the singularity curve on some set $S$ is
introduced for generality of the discussion. Hereafter, we say that a
parametric curve $\\{(\theta(s),I(s))\\}_{s\in S^{1}}$ is a dividing circle if
$\theta(s),I(s)$ are continuous and periodic in $s$, the curve is not self
intersecting, and $\\{\theta(s)\\}_{s\in S^{1}}=[-\pi,\pi]$. For $\epsilon=0$,
all the circles with a fixed $I$ are invariant and a tangency occurs if and
only if $I=I_{tan}(E)$. Thus the unperturbed parametric singularity curve is
identical to the singularity set and is given by the circle:
$\sigma^{0}=\mathcal{\bar{S}}^{0}=\\{(\theta_{tan}^{0},I_{tan}^{0})|I_{tan}^{0}(s)=I_{tan}(E),\theta_{tan}^{0}(s)=s,s\in[-\pi,\pi]\\}$,
namely, at $\epsilon=0$, the tangency curve is a graph. In section 3 we
establish that for sufficiently small $\epsilon$ the perturbed singularity
curve and its first image are also smooth graphs:

######  Theorem 1.

For $E>V_{1}(q_{1}^{w})$ and sufficiently small $\epsilon$, the singularity
curve $\sigma^{\epsilon}(E)$ of the wall system is an isolated dividing
circle, dividing the cylinder $\Sigma_{E}$ to impacting (below the curve) and
non-impacting (above the curve) regions of the first return map,
$\mathcal{F}_{\epsilon}$. Moreover, this circle is a graph:
$\sigma^{\epsilon}=\\{(\theta,I_{tan}^{\epsilon}(\theta)),\theta\in[-\pi,\pi]\\}$,
its image under the map is also a graph,
$\mathcal{F}_{\epsilon}\sigma^{\epsilon}=\\{(\theta,\bar{I}_{tan}^{\epsilon}(\theta)),\theta\in[-\pi,\pi]\\}$
and both $I_{tan}^{\epsilon}(\theta)$ and $\bar{I}_{tan}^{\epsilon}(\theta)$
are $C^{r}$ $\epsilon-$close to the constant function $I_{tan}(E)$. Finally,
the singularity curve and its image are related by the reflection symmetry:
$\bar{I}_{tan}^{\epsilon}(\theta)=I_{tan}^{\epsilon}(-\theta)$.

Next, we study the extent of the singularity set. By Corollary 3.2 in [23],
for any $\rho>0$, for sufficiently small $\epsilon$, there are KAM curves that
remain at a distance $\rho$ from $\sigma^{0}$: the family of KAM curves that
are $\rho$-below $\sigma^{0}$ consist of quasi-periodic orbits with transverse
impacts whereas those KAM tori which are $\rho$-above $\sigma^{0}$ consist of
KAM curves of the smooth system that remain bounded away from the wall. We
thus define:

######  Definition 2.5.

The _tangency band_ , $\mathcal{B}^{\epsilon}$, is the smallest closed band, a
set enclosed in between two dividing circles, which includes
$\mathcal{\bar{S}}^{\epsilon}$. The lower boundary of $\mathcal{B}^{\epsilon}$
is the impacting dividing circle,
$\\{(I^{\epsilon}_{-}(s),\theta^{\epsilon}_{-}(s))\\}_{s\in S^{1}}$ and its
upper boundary is the non-impacting dividing circle
$\\{(I^{\epsilon}_{+}(s),\theta^{\epsilon}_{+}(s))\\}_{s\in S^{1}}$.

The maximal width of the band is
$W=\max_{\theta}(I^{\epsilon}_{+}(s)|_{\theta^{\epsilon}_{+}(s)=\theta}-I^{\epsilon}_{-}(s)|_{\theta^{\epsilon}_{-}(s)=\theta})$,
and its upper and lower maximal widths are
$W^{+}=\max_{\theta}(I^{\epsilon}_{+}(s)|_{\theta^{\epsilon}_{+}(s)=\theta}-I^{\epsilon}_{tan}(\theta))$
and
$W^{-}=\max_{\theta}(I^{\epsilon}_{-}(s)|_{\theta^{\epsilon}_{-}(s)=\theta}-I^{\epsilon}_{tan}(\theta))$
respectively so $W^{\pm}\leqslant W\leqslant W^{+}+W^{-}$.

Corollary 3.2 in [23] implies that for $\alpha<\frac{1}{2}$, for sufficiently
small $\epsilon$, there exist a constant $C$ (depending on the parameters and
energy) such that
$\mathcal{B}^{\epsilon}\subset\\{(\theta,I)|I\in[I_{tan}(E)-C\epsilon^{\alpha},I_{tan}(E)+C\epsilon^{\alpha}],\theta\in[-\pi,\pi]\\}$,
namely, one can take $\rho$ to approach zero slower than $\sqrt{\epsilon}$ to
establish the persistence of bounding KAM curves which are at a distance
$\rho$ from the singularity line. Thus, there exists $C>0$ such that
$W\leqslant C\epsilon^{\alpha}$ for all $\alpha<\frac{1}{2}$. Here, we
construct and study $\mathcal{F}_{\epsilon}$ in the singularity band and
establish tighter bounds on the upper and lower singularity band widths.

First, we establish that the division of $\Sigma_{E}$ to impacting and non-
impacting initial conditions allows to approximate the return map in each part
by employing perturbation methods, leading to the main analytical result of
this paper, established in section 4:

######  Theorem 2.

For $E>V_{1}(q_{1}^{w})$, for sufficiently small $\epsilon$, the return map
$\mathcal{F}_{\epsilon}:\Sigma_{E}\rightarrow\Sigma_{E}$ of the wall system
flow near the singularity set can be brought, by a $C^{r}$ smooth symplectic
change of coordinates, to the symplectic form:

$\mathcal{F}^{loc}_{\epsilon}:\begin{cases}\bar{K}=K+\epsilon
f(\bar{\phi})+\mathcal{O}_{C^{r-2}}(\epsilon G_{s}(K))\\\
\bar{\phi}=\phi+\Omega+G_{s}(K)+\mathcal{O}_{C^{r-2}}(\epsilon,\epsilon
G_{s}(K),K^{2})\end{cases}$ (9)

where $\epsilon
f(\bar{\phi})=\bar{I}^{\epsilon}_{tan}(\bar{\phi})-I^{\epsilon}_{tan}(\bar{\phi})=\epsilon\int_{0}^{T^{tan}_{1}(E)}\\{I,V_{c}\\}\mid_{z_{tan}(t,\theta;E)}dt+\mathcal{O}_{C^{r}}(\epsilon^{2})$
is the difference between the pre-image and the forward image of the tangency
curve at $\bar{\phi}$,
$\Omega=\Omega(E)=2\pi\frac{T_{1}(V_{1}(q_{1}^{w}))}{T_{2}(E-V_{1}(q_{1}^{w}))}=\Theta(I_{tan}(E);E)$
is the unperturbed rotation number at the unperturbed tangent action (4),
$G_{s}(K)$ is a continuous piecewise smooth function with a square root
singularity at the origin:

$G_{s}(K)=\begin{cases}\tau(I_{tan}(E))\cdot
K+\mathcal{O}_{C^{r}}(K^{2})&K\geq 0\\\
-\frac{(2\omega_{2}(I_{tan}(E)))^{3/2}}{|V_{1}^{\prime}(q_{1}^{w})|}\cdot\sqrt{-K}+\tau(I_{tan}(E))\cdot
K+\mathcal{O}_{C^{r-2}}((-K)^{3/2},K^{2})&K<0\end{cases}$ (10)

and

$\tau(I)=\frac{d}{dI}\left(\omega_{2}(I)\cdot T_{1}(E-H_{2}(I))\right)$ (11)

is the twist of the return map of the unperturbed smooth Hamiltonian system.

By $\mathcal{O}_{C^{r}}(a(\epsilon),b(K),\epsilon G_{s}(K))$ we mean that for
sufficiently small $\epsilon,|K|$ this term may be represented by a $C^{r}$
smooth function of its arguments, $g(a(\epsilon),b(K),\epsilon G_{s}(K))$,
satisfying $g(0,0,0)=0$. Namely, this term is small and its $j$-th derivative
w.r.t. $K$ can grow at most as $\epsilon^{j}G_{s}^{(j)}(K)$ (and similar
growth estimates apply to the singular terms in $G_{s}(K)$). The coordinates
$(K,\phi)$ are related to the $(I,\theta)$ coordinates by a $\theta$-dependent
shift of the singularity action, so that for all small $\epsilon$ the
singularity curve in these coordinates is the circle $K=0,\phi\in[-\pi,\pi]$,
impacts correspond to negative $K$, the region near the singularity line
corresponds to small $|K|$ values, and the upper and lower widths of the
singularity band in the $K$ variable are simply the maximal positive and
minimal negative $K$ values of the singularity set.

In Theorem 5.1 we establish that for sufficiently small $\epsilon$, the map
$\mathcal{F}^{loc}_{\epsilon}$ has invariant KAM curves that remain, for
sufficiently small $\epsilon$, at a signed distance
$\pm\epsilon^{\gamma_{\pm}(\Omega)}$ away from the singularity line $K=0$;
namely, there exist constants $\gamma_{\pm}(\Omega)>0$ and constants
$k_{\pm}(\Omega,\gamma_{\pm}(\Omega)),\bar{k}_{\pm}(\Omega,\gamma_{\pm}(\Omega))>0$,
such that $\mathcal{F}^{loc}_{\epsilon}$ has KAM curves
$(K_{\pm}(\phi),\phi)$, satisfying
$k_{+}\epsilon^{\gamma_{+}}<K_{+}(\phi)<\bar{k}_{+}\epsilon^{\gamma_{+}}$ and
$-\bar{k}_{-}\epsilon^{\gamma_{-}}<K_{-}(\phi)<-k_{-}\epsilon^{\gamma_{-}}$.
For Diophantine $\Omega$ values $\gamma_{\pm}(\Omega)\lesssim 1$ whereas for
resonant $\Omega$ values $\gamma_{+}(\Omega)\lesssim 1/2$ and
$\gamma_{-}(\Omega)\lesssim 2/3$. These provide a refinement of the results of
Corollary 3.2 in [23] which were obtained by analysing the map
$\mathcal{F}_{\epsilon}$ away from the tangency zone. This implies:

######  Theorem 3.

For $E>V_{1}(q_{1}^{w})$, for sufficiently small $\epsilon$, the width of the
tangency band for a Diophantine $\Omega$ value is of $\mathcal{O}(\epsilon)$
whereas for resonant values of $\Omega$ the tangency band is asymmetric, with
its upper width being at most of order $\mathcal{O}(\sqrt{\epsilon})$ and its
lower width being at most of $\mathcal{O}(\epsilon^{2/3})$.

To examine whether these bounds are realized and to study the dynamics in the
tangency band we examine in section 6 a truncated version of the map (9), the
truncated tangency map $F_{TTM}$, in which we set $g_{1}=g_{2}=0$:

$F_{TTM}:\begin{cases}\bar{K}=K+\epsilon f(\bar{\phi})\\\
\bar{\phi}=\phi+\Omega+\tau\cdot K-\alpha\sqrt{-K}\cdot Heav(-K)\end{cases}$
(12)

$Heav(K)$ denotes the heaviside function, and by re-scaling $K$ we take
$\tau\in\\{1,-1\\}$. After establishing some basic properties of this map
(fixed points and twist), we examine numerically, for
$f(\bar{\phi})=\sin(\bar{\phi})$ (i.e. for the first Fourier mode of $f)$, the
width and structure of the tangency band of the truncated map. We observe that
the widths scaling with $\epsilon$ near $K=0$ for resonant and Diophantine
$\Omega$ are very similar as long as $\Omega$ is not close to the strongest
resonance cases, $\Omega\approx 2\pi\text{ (mod $2\pi$)}$. So we call the non-
strong-resonance cases general-$\Omega$ cases. For the strong resonance cases,
the map (12) has fixed points which are close to the singularity line,
undergoing different types of bifurcations in the positive shear ($\tau=1$)
and the negative shear ($\tau=-1$) cases. Thus, we divide our numerical
observations to three cases: general-$\Omega$ , positive shear strong
resonance and negative shear strong resonance cases:

######  Claim 2.6.

For the general-$\Omega$ cases the tangency band width is of
$\mathcal{O}(\epsilon)$, whereas for the strong resonance cases its upper
width is of order $\mathcal{O}(\sqrt{\epsilon})$ and its lower width is of
$\mathcal{O}(\epsilon^{2/3})$.

######  Claim 2.7.

The set $\mathcal{B}^{\epsilon}\setminus\mathcal{\bar{S}}^{\epsilon}$ may
include two sided islands: stable periodic orbits surrounded by quasi-periodic
orbits that hop between impacting and non-impacting returns to $\Sigma_{E}$.

######  Claim 2.8.

For the general-$\Omega$ cases and for the positive shear strong resonance
case, the tangency band includes a connecting orbit which visits arbitrarily
close the upper and lower boundaries of $\mathcal{B}^{\epsilon}$, yet, the
connecting time, $N_{c}(\epsilon,\alpha,\Omega,\tau)$, diverges as $\epsilon$
and/or $\alpha$ are decreased.

The implication of these large transient connection times is that for very
long periods (e.g., for $\epsilon=0.01$ and $\alpha=1$, for more than $10^{7}$
iterates) the density of orbits is non-uniform. During these transient times
orbits cover essentially densely overlapping yet different parts of the same
ergodic component. Interestingly, for the negative shear strong resonance
case, at $\Omega$ values which are slightly below $2\pi k$, there are $\alpha$
values for which no such connecting orbits are found, and, some orbits
belonging to the singular set do not overlap even after $10^{10}$ iterates. We
conclude that even if there is a connecting orbit,
$N_{c}(\epsilon,\alpha,2\pi,-1)$ is much larger than all other cases. Whether
such a connecting orbit exists, or, alternatively, the singularity set is
divided to several invariant subsets for this case, remains an open question.

##  3 The singularity curve

Recall that in the unperturbed setting, the iso-energy singularity curve,
$\sigma^{0}(E)\subset\Sigma_{E}$, is the invariant circle
$\sigma^{0}(E)=\\{(\theta,I_{tan}(E))$, $\theta\in[-\pi,\pi]\\}$, and it
separates $\Sigma_{E}$ to impacting ($I<I_{tan}(E)$) and non-impacting
($I>I_{tan}(E)$) circles. Here we prove Theorem 1 which asserts that for
sufficiently small $\epsilon>0$ the singularity curve $\sigma^{\epsilon}(E)$
(which is generically not invariant) has similar dividing properties and is a
smooth graph. We first characterize the curve which divides between impacting
and non-impacting points at $\Sigma_{E}^{*}$, a properly defined section near
the wall, and then integrate this wall tangent curve backward and forward to
$\Sigma_{E}$ to establish the properties of the singularity curve and its
image.

###  3.1 Characterization of tangency at a section near the wall

Define an auxiliary iso-energy cross-section $\Sigma_{E}^{*}$ near the wall
(see Figure 1):

######  Definition 3.1.

The iso-energy star-section $\Sigma_{E}^{*}$ is the union of the inward impact
wall section: $\Sigma_{E}^{w}=\\{(q,p)|H(q,p)=E,q_{1}=q_{1}^{w},p_{1}\leqslant
0\\}$ and the by-pass section
$\Sigma_{E}^{>}=\\{(q,p)|H(q,p)=E,p_{1}=0,\dot{p}_{1}>0,q_{1}\geq
q_{1}^{w}\\}$, namely $\Sigma_{E}^{*}=\Sigma_{E}^{>}\cup\Sigma_{E}^{w}$. The
auxiliary section (which may be crossed only by the auxiliary smooth flow) is
the section
$\Sigma_{E}^{<}=\\{(q,p)|H(q,p)=E,p_{1}=0,\dot{p}_{1}>0,q_{1}<q_{1}^{w}\\}$.

Figure 1: Near tangent dynamics in the $(q_{1},p_{1})$ plane for a single
iteration of the unperturbed return map $\Sigma_{E}\rightarrow\Sigma_{E}$.
Filled circles - indicate i.c., circles - position at the wall, diamonds - end
points, i.e. the image of the return map. The phase $\theta$ gained by the
impact flow (red diamond) is smaller than the phase gained by the smooth flow
(black diamond). Here $q_{1}^{w}=-0.8$,
$V_{1}(q_{1})=2q_{1}^{2}+q_{1}^{3}+\frac{1}{4}q_{1}^{4}$,
$V_{2}(q_{2})=\frac{3}{2}q_{2}^{2},H=2$ and $\epsilon=0$.

For all $E>0$, the piecewise smooth two dimensional star section
$\Sigma_{E}^{*}$ is well defined and is non-empty. For $E<V_{1}(q_{1}^{w})$
and sufficiently small $\epsilon$, the wall section, $\Sigma_{E}^{w}$ is the
empty set and the star section consists of the by-pass section alone: for such
energies all segments by-pass the wall without an impact. For
$E>V_{1}(q_{1}^{w})$ both $\Sigma_{E}^{>},\Sigma_{E}^{w}$ are non-empty and
include the wall iso-energy tangent curve:

######  Lemma 3.2.

The collection of all tangential wall positions with energy
$E>V_{1}(q_{1}^{w})$ is given by:

$\Xi_{wall}^{\epsilon}(E)=\\{(q_{1}=q_{1}^{w},p^{w}_{1}=0,q_{2},p_{2})|E-V_{1}(q_{1}^{w})=\frac{p_{2}^{2}}{2}+V_{2}(q_{2})+\epsilon\cdot
V_{c}(q_{1}^{w},q_{2})\\}.$ (13)

For sufficiently small $\epsilon$ the projection $\Pi$ of this curve to the
$(q_{2},p_{2})$ plane, the iso-energy wall tangency curve,
$\sigma_{wall}^{\epsilon}(E)=\Pi\Xi_{wall}^{\epsilon}(E)$, is a circle which
can be represented as a graph in the unperturbed action angle coordinates
$(I,\theta)$:

$\sigma_{wall}^{\epsilon}(E)=\\{(\theta,I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta))|\theta\in[-\pi,\pi]\\}$
(14)

where
$I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)=I_{tan}(E)+O_{C^{r}}(\epsilon)$ is
a $C^{r}$-smooth function.

###### Proof.

Since the wall is the vertical line $q_{1}=q_{1}^{w}$, a tangent wall position
occurs if and only if $q_{1}=q_{1}^{w},p^{w}_{1}=0$, and (13) follows from
plugging these conditions in the perturbed Hamiltonian (1). The obtained curve
is a circle: from the S3B property, the potential $V_{2}(q_{2})$ has bounded
level sets, so, for all $H>V_{1}(q_{1}^{w}),$ at $\epsilon=0$, Eq. (13)
defines a circle in the $(q_{2},p_{2})$ plane:

$\\{(q_{2},p_{2})|p_{2}=p_{2}^{tan,\pm,0}(q_{2};H)=\pm\sqrt{2\left(H-V_{1}(q_{1}^{w})-V_{2}(q_{2})\right)},q_{2}\in[q_{2min}^{0}(E),q_{2max}^{0}(E)]\\}.$
(15)

Defining
$(I_{tan,\Sigma^{*}_{E}}^{0}(q_{2},p_{2}),\theta^{0}_{tan,\Sigma^{*}_{E}}(q_{2},p_{2}))=S_{2}(q_{2},p_{2}^{tan,\pm,0}(q_{2};E)),$
the unperturbed singularity curve at the wall representation in the action
angle coordinates is $I_{tan,\Sigma^{*}_{E}}^{0}(q_{2},p_{2})=I_{tan}(E)$ and
$\theta^{0}_{tan,\Sigma^{*}_{E}}\in[-\pi,\pi]$. In particular, the constant
function
$I_{tan,\Sigma^{*}_{E}}^{0}(q_{2},p_{2})=I^{0}_{tan,\Sigma^{*}_{E}}(\theta^{0}_{tan,\Sigma^{*}_{E}}(q_{2},p_{2}^{tan,\pm,0}(q_{2};E)))$
is a graph of $\theta^{0}_{tan,\Sigma^{*}_{E}}$, satisfying
$\frac{dI^{0}_{tan,\Sigma^{*}_{E}}(\theta)}{d\theta}=0.$

For all $H>V_{1}(q_{1}^{w})$, for $\epsilon$ sufficiently small, this property
is preserved by $V_{2}(q_{2})+\epsilon V_{c}(q_{1}^{w},q_{2})$; away from
$q_{2min,2max}^{0}(E)$ the solution to (13) depends $C^{r+1}$ \- smoothly on
$\epsilon$ and is of the form
$(q_{2},p_{2}^{tan,\pm,\epsilon}(q_{2};E),q_{2})$:

$p_{2}^{tan,\pm,\epsilon}(q_{2};E)=\pm\sqrt{2(E-V_{1}(q_{1}^{w})-V_{2}(q_{2})-\epsilon\cdot
V_{c}(q_{1}^{w},q_{2}))}.$ (16)

Close to the turning points, since $V_{2}^{\prime}(q_{2min/max}^{0}(E))\neq
0$, the solution to (13) depends $C^{r+1}$ \- smoothly on $\epsilon$ and is of
the form $(q_{2}^{tan,\pm,\epsilon}(p_{2};E),p_{2})$: in particular, there
exist $q_{2min/max}^{\epsilon}(E)=q_{2}^{tan,\pm,\epsilon}(0;E)\approx
q_{2min/max}^{0}(E)+O_{C^{r}}(\epsilon)$ such that
$\Xi_{wall}^{\epsilon}(E)=\\{(q_{1}^{w},0,q_{2},p_{2})|p_{2}=p_{2}^{tan,\pm,\epsilon}(q_{2};E),q_{2}\in[q_{2min}^{\epsilon}(E),q_{2max}^{\epsilon}(E)]\\}$,
so indeed we have a circle (both $\sigma_{wall}^{\epsilon}(E)$ in the
$(q_{2},p_{2})$ plane and $\Xi_{wall}^{\epsilon}(E)$ in the full phase space).
Defining
$(\theta_{tan,\Sigma^{*}_{E}}^{\epsilon}(q_{2}),I_{tan,\Sigma^{*}_{E}}^{\epsilon}(q_{2}))=S_{2}(q_{2},p_{2}^{tan,\pm,\epsilon}(q_{2};E))$
in the chart of $q_{2}$ values bounded away from the turning points and
$(,\theta_{tan,\Sigma^{*}_{E}}^{\epsilon}(p_{2}),I_{tan,\Sigma^{*}_{E}}^{\epsilon}(p_{2}))=S_{2}(q_{2}^{tan,\pm,\epsilon}(p_{2};E),p_{2})$
at the chart near the turning points establishes the existence of a $C^{r}$
smooth action angle presentation of the tangent circle (14) for all
$E>V_{1}(q_{1}^{w})$ and sufficiently small $\epsilon$, and its $C^{r}$
closeness to the circle $(\theta,I_{tan,\Sigma^{*}_{E}}^{0}(\theta))$. In
particular, in the first chart
$\frac{dI_{tan,\Sigma^{*}_{E}}^{\epsilon}(q_{2})}{d\theta_{tan,\Sigma^{*}_{E}}^{\epsilon}(q_{2})}=\frac{dI_{tan,\Sigma^{*}_{E}}^{\epsilon}(q_{2})/dq_{2}}{d\theta_{tan,\Sigma^{*}_{E}}^{\epsilon}(q_{2})/dq_{2}}=\frac{O_{C^{r-1}}(\epsilon)}{d\theta_{tan,\Sigma^{*}_{E}}^{0}(q_{2})/dq_{2}+O_{C^{r-1}}(\epsilon)}=O_{C^{r-1}}(\epsilon)$
(since away from the turning points
$d\theta_{tan,\Sigma^{*}_{E}}^{0}(q_{2})/dq_{2}$ is bounded away from $0$),
and similar calculation applies to the second chart near the turning points.
It thus follows that the circle can be represented as a graph in the
unperturbed action angle coordinates representation.

∎

Next, we establish that the wall tangency circle divides the plane
$\Pi\Sigma_{E}^{*}$ to impacting and non-impacting trajectory segments:

######  Lemma 3.3.

For sufficiently small $\epsilon$ the $(q_{2},p_{2})$ coordinates parametrize
$\Sigma^{*}_{E}$: to each point $(q_{2},p_{2})\in\Pi\Sigma^{*}_{E}$ there
exists a unique point $z\in\Sigma^{*}_{E}$. The wall tangency circle,
$\sigma_{wall}^{\epsilon}(E)$, divides $\Pi\Sigma^{*}_{E}$ between iso-energy
impacting (interior points to $\sigma_{wall}^{\epsilon}(E)$ corresponding to
$z\in\Sigma^{w}_{E}$) and non-impacting points (exterior points to
$\sigma_{wall}^{\epsilon}(E)$, corresponding to $z\in\Sigma^{>}_{E}$) on
$\Sigma_{E}^{*}$.

###### Proof.

A trajectory impacts if and only if it has sufficient kinetic energy in the
$q_{1}$ direction to reach the wall $q_{1}=q_{1}^{w}$. In other words, for a
given energy $E$, for a given $(q_{2},p_{2})$, impact happens iff:

$\frac{p_{1}^{2}}{2}\overset{at\
wall}{=}E-V_{1}(q_{1}^{w})-\frac{p_{2}^{2}}{2}-V_{2}(q_{2})-\epsilon
V_{c}(q_{1}^{w},q_{2})\geq 0$ (17)

Or equivalently

$\frac{p_{2}^{2}}{2}+V_{2}(q_{2})+\epsilon V_{c}(q_{1}^{w},q_{2})\leq
E-V_{1}(q_{1}^{w})$ (18)

In this case,
$z^{w}=(q_{1}^{w},p_{1}^{w}=-p^{\epsilon}_{1}(q_{2},p_{2},E)=-\sqrt{2(E-V_{1}(q_{1}^{w})-\frac{p_{2}^{2}}{2}-V_{2}(q_{2})-\epsilon
V_{c}(q_{1}^{w},q_{2}))},q_{2},p_{2})\in\Sigma_{E}^{w}$ is the unique
intersection point with $\Sigma_{E}^{*}$, and the case of equality in (18)
sets $p_{1}=0$, so for this case only, this point also belongs to
$\Sigma_{E}^{>}$. It follows from (18) that in the $(q_{2},p_{2})$ plane such
points are inside (and respectively, on) the iso-energy tangency curve.

On the other hand, consider a $(q_{2},p_{2})$ point which is outside the
tangency curve, namely, there exists some $\eta>0$ such that:

$\frac{p_{2}^{2}}{2}+V_{2}(q_{2})+\epsilon
V_{c}(q_{1}^{w},q_{2})=E-V_{1}(q_{1}^{w})+\eta.$ (19)

We need to show that for such a point there exist a unique
$q_{1,min}^{\epsilon,\eta}\in(q_{1}^{w},0)$ such that
$(q_{1,min}^{\epsilon,\eta},p_{1}=0,q_{2},p_{2})\in\Sigma_{E}^{>}$. By the S3B
assumption, for any $\eta\in(0,V_{1}(q_{1}^{w}))$ there exists
$q_{1,min}^{0,\eta}\in(q_{1}^{w},0)$ such that
$E-V_{1}(q_{1,min}^{0,\eta})=E-V_{1}(q_{1}^{w})+\eta$, i.e.
$V_{1}(q_{1,min}^{0,\eta})=V_{1}(q_{1}^{w})-\eta$ and
$V_{1}^{\prime}(q_{1,min}^{0,\eta})<0$. Thus, by the inverse function theorem,
for $\epsilon$ sufficiently small, there exists a unique
$q_{1,min}^{\epsilon,\eta}=q_{1,min}^{0,\eta}+O(\epsilon)$ for which:

$\frac{p_{2}^{2}}{2}+V_{2}(q_{2})+\epsilon
V_{c}(q_{1,min}^{\epsilon,\eta},q_{2})=E-V_{1}(q_{1,min}^{\epsilon,\eta}).$
(20)

For any fixed $\eta>0$, taking $\epsilon=o(\eta)$ sufficiently small, we
conclude that $q_{1,min}^{\epsilon,\eta}\in(q_{1}^{w},0)$ and indeed
$(q_{1,min}^{\epsilon,\eta},p_{1}=0,q_{2},p_{2})\in\Sigma_{E}^{>}$. Now, we
need to show that for $(q_{2},p_{2)})$ values which satisfy (19) with
arbitrary small positive $\eta$ the same conclusion applies, namely that the
solution to (20) satisfies $q_{1,min}^{\epsilon,\eta}\in(q_{1}^{w},0)$.
Indeed, subtracting the two equations we obtain

$\eta(q_{1,min}^{\epsilon,\eta},q_{2})=V_{1}(q_{1}^{w})-V_{1}(q_{1,min}^{\epsilon,\eta})+\epsilon(V_{c}(q_{1}^{w},q_{2})-V_{c}(q_{1,min}^{\epsilon,\eta},q_{2}))$
(21)

so $\eta$ can vanish only if $V_{1}^{\prime}(q_{1}^{w})=O(\epsilon)$ which is
false by the S3B assumption and the assumption that $q_{1}^{w}<0$. Since for
small $\eta=\eta(\epsilon)$,
$q_{1,min}^{0,\eta}=q_{1}^{w}+\frac{\eta}{|V_{1}^{\prime}(q_{1}^{w})|}+O(\eta^{2})$,
we conclude that $q_{1,min}^{\epsilon,\eta}=q_{1}^{w}+\bar{q}_{1}$ where
$\bar{q}_{1}=O(\epsilon,\eta)$. Plugging in this expression in (20) and using
(19), we find that for arbitrary small positive $\eta$, the next order term,
$\bar{q}_{1}$, must be positive:

$\bar{q}_{1}=\frac{-\eta}{V_{1}^{\prime}(q_{1}^{w})}+O_{C^{r-1}}(\epsilon\bar{q}_{1}^{2})+O_{C^{r-1}}(\bar{q}_{1}^{2})+O_{C^{r}}(\epsilon\eta)>0$
(22)

so indeed, the point
$z=(q_{1,min}^{\epsilon,\eta},p_{1}=0,q_{2},p_{2})\in\Sigma_{E}^{>}$. ∎

###  3.2 Forward and backward flow from the near wall section.

We establish next that $\Sigma_{E}^{*}$ is crossed exactly once before
returning to $\Sigma_{E}$ and that the crossing times are well defined and
have a continuous limit as $\epsilon\rightarrow 0$:

######  Lemma 3.4.

For all $E>0$ and sufficiently small $\epsilon$, the wall flow of an initial
condition $z_{0}\in\Sigma_{E}$ crosses the star section exactly once before
returning to $\Sigma_{E}$: for any $z\in\Sigma_{E}$ there exists a unique
star-crossing time, $t^{-,\epsilon}(z_{0})$, such that
$\Phi^{\epsilon,im}_{t^{-,\epsilon}(z_{0})}z_{0}\in\Sigma_{E}^{*}$ and a
unique return time,
$\tilde{T}_{1}^{\epsilon}(z_{0})=t^{-,\epsilon}(z_{0})+t^{+,\epsilon}(z_{0})$
such that
$\Phi^{\epsilon,im}_{\tilde{T}_{1}^{\epsilon}(z_{0})}z_{0}\in\Sigma_{E}$.
Moreover, $\lim_{\epsilon\rightarrow
0}t^{\pm,\epsilon}(z_{0})=\frac{1}{2}\tilde{T}_{1}(J(I(q_{20},p_{20}),E))$.

###### Proof.

Denote by $z=(q_{10},p_{10}=0,q_{20},p_{20})\in\Sigma_{E}$ where $q_{10}>0$.
Then, since $\dot{q}_{1}=p_{1},\
\dot{p}_{1}|_{\Sigma_{E}}=-V_{1}^{\prime}(q_{1max}(E))+O(\epsilon)<0$, $q_{1}$
decreases until either $p_{1}$ changes its sign (so it crosses $\Sigma^{>}$,
possibly at $q_{1}^{w}$) or it reaches (possibly tangentially) the wall,
crossing $\Sigma^{w}$. Namely, there exists a unique first crossing time of
the star section, $t^{-,\epsilon}(z_{0})$. In the first case the trajectory
continues without impact whereas in the second case it impacts the wall, so
$p_{1}(t^{-,\epsilon}(z_{0}))\rightarrow-p_{1}(t^{-,\epsilon}(z_{0}))\geq 0$.
In both cases, just after crossing $\Sigma^{*}_{E}$, the horizontal velocity
strictly increases and is/becomes positive (since either
$\dot{p}_{1}|_{\Sigma^{>}_{E}}=-V_{1}^{\prime}(q_{1min}(E))+O(\epsilon)>0$ or
$\dot{p}_{1}|_{\Sigma^{w}_{E}}=-V_{1}^{\prime}(q^{wall}_{1})+O(\epsilon)>0$).
This trend reverses once $q_{1}$ becomes positive, leading to the return to
$\Sigma_{E}$ at a unique return time
$\tilde{T}_{1}^{\epsilon}(z_{0})=t^{-,\epsilon}(z_{0})+t^{+,\epsilon}(z_{0})$.
The limit of the return time follows from the smooth dependence of the flow on
$\epsilon$ in between impacts and the symmetry in $p_{1}$ of the unperturbed
wall flow.∎

The flow restricted to i.c. which cross through the by-pass section, and in
particular, to the i.c. belonging to the singularity curve, is identical to
the smooth perturbed flow which is $C^{r}$ $\epsilon-$close to the unperturbed
flow. On the other hand, the dependence of the return times on initial
conditions is non-smooth near tangency for impacting initial conditions, i.e.
for i.c. that cross $\Sigma^{w}$. In section 4.1 we derive the expansion for
the non-smooth part. The tangency curve, by definition, belongs to the common
boundary of $\Sigma_{E}^{>},\Sigma^{w}_{E}$, namely,
$\Xi_{wall}^{\epsilon}(E)\subset(\Sigma_{E}^{>}\cap\Sigma_{E}^{w})\subset\Sigma_{E}^{*}$
and thus:

######  Lemma 3.5.

For $E>V_{1}(q_{1}^{w})$ and for sufficiently small $\epsilon$, the iso-energy
singularity curve of the wall system can be represented as a $C^{r}$ smooth
graph in the unperturbed action-angle coordinates:
$\sigma^{\epsilon}(E)=\\{(\theta,I_{tan}^{\epsilon}(\theta;E)),\theta\in[-\pi,\pi]\\}$.

###### Proof.

First, notice that the singularity curve $\sigma^{\epsilon}(E)$ is the
projection to the $(q_{2},p_{2})$ plane of the backwards flow of
$\Xi_{wall}^{\epsilon}(E)$ to $\Sigma_{E}$. Indeed, denote by
$z_{0}=(q_{10},p_{10}=0,q_{20},p_{20})\in\Xi^{\epsilon}(E)\subset\Sigma_{E}$
an initial condition with $(q_{20},p_{20})$ belonging to
$\sigma^{\epsilon}(E)$ (since $\Sigma_{E}$ is a transverse section
$q_{10}=q_{10}(q_{20},p_{20},E)$ is uniquely defined). Then, by lemma 3.4,
there exists a unique crossing time of $t^{-,\epsilon}(z_{0})$ of
$\Sigma^{*}_{E}$, and, by definition of the singularity curve the crossing of
$\Sigma^{*}_{E}$ must be tangent to the wall, so indeed
$z(t^{-,\epsilon}(z_{0}))\in\Xi_{wall}^{\epsilon}(E)$. Conversely, let
$z^{w}=(q_{1}^{w},p^{w}_{1}=0,q_{2},p_{2}=p_{2}^{tan,\pm,\epsilon}(q_{2};H))\in\Xi_{wall}^{\epsilon}(E)$.
Then, integrating backwards the smooth flow from this point, and noticing, as
in lemma 3.4, that
$\dot{p}_{1}|_{\Sigma^{w}_{E}}=-V_{1}^{\prime}(q^{wall}_{1})+O(\epsilon)>0$
(since $q_{1}^{wall}<0$), we obtain that there exist a unique first backward
crossing time of the section $\Sigma_{E}$. By uniqueness of solutions we
conclude that for every $z^{w}\in\Xi_{wall}^{\epsilon}(E)$ there exists a
unique $z_{0}(z_{w})\in\Xi^{\epsilon}(E)$ such that
$z^{w}=\Phi^{\epsilon}_{t^{-,\epsilon}(z_{0})}z_{0}$ and thus
$z_{0}=\Phi^{\epsilon}_{-t^{-,\epsilon}(z_{0})}z^{w}$. Since by lemma 3.2 the
tangency curve $\Xi_{wall}^{\epsilon}(E)$ is a smooth circle, and the backward
crossing time depends smoothly on i.c. belonging to
$\Xi_{wall}^{\epsilon}(E)$, its pre-image on $\Sigma_{E}$ by the smooth
diffeomorphism is also a smooth circle.

Using the unperturbed transformation to action angle coordinates, the backward
evolution of $\Xi_{wall}^{\epsilon}(E)$ from $\Sigma_{E}^{>}$ to the section
$\Sigma_{E}$ is represented, in the $(q_{2},p_{2})$ plane by the map
$\\{(\theta,I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)),\theta\in[-\pi,\pi]\\}\rightarrow\\{(\theta^{\prime}(\theta;E),I^{\prime}(\theta;E))|\theta\in[-\pi,\pi]\\}=\sigma^{\epsilon}(E)\subset\Sigma_{E}$,
where:

$\begin{cases}I^{\prime}(\theta;E)-I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)&=\epsilon
f^{-}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta),\theta;\epsilon)\\\
\theta^{\prime}(\theta;E)-\theta&=-\frac{1}{2}\cdot
T_{1}(E,I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta))\cdot\omega_{2}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta))+\epsilon
g^{-}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta),\theta;\epsilon).\end{cases}$
(23)

The right hand sides correspond to the backward integrals of the perturbed
smooth vector field
$(\frac{dI}{d(-t)}=-\epsilon\\{I,V_{c}\\},\frac{d\theta}{d(-t)}=-\\{\theta,H\\}=-\omega_{2}(I)-\epsilon\\{\theta,V_{c}\\})$
along the smooth perturbed trajectory. Since tangent trajectories do not
impact, their evolution under the perturbed smooth flow is $C^{r}$-close to
the integrable motion, so the right hand side in (23) and the $f^{-},g^{-}$
perturbation terms are smooth in $\theta$ and $\epsilon$ and $f^{-}$ may be
found to leading order by a Melnikov-type calculation (see section 4). Thus,
the singularity curve is given parametrically by
$\sigma^{\epsilon}(E)=\\{(\theta^{\prime}(\theta;E),I^{\prime}(\theta;E))|\theta\in[-\pi,\pi]\\}$.
To complete the proof we need to show that it is a graph in the angle
coordinate, $\theta^{\prime}$, namely, we need to show that
$\frac{d\theta^{\prime}}{d\theta}\neq 0$. Now,

$\begin{split}\frac{d\theta^{\prime}}{d\theta}&=1-\frac{1}{2}\frac{d}{d\theta}\left(T_{1}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta))\cdot\omega_{2}(I_{tan,\Sigma_{1}}^{\epsilon}(\theta))\right)+\epsilon\frac{dg^{-}}{d\theta}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta),\theta;\epsilon)\\\
&=1-\frac{1}{2}\tau(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta))\frac{dI_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)}{d\theta}+\epsilon\frac{dg^{-}}{d\theta}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta),\theta;\epsilon)\end{split}$
(24)

where $\tau(I)$ is the twist (see Eq. (11)) of the return map for the smooth
Hamiltonian system. By Lemma 3.2 (and in particular, as it implies that
$\frac{dI_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)}{d\theta}=O_{C^{r-1}}(\epsilon)$
), and since the unperturbed twist, $\tau(I)$, is bounded, for sufficiently
small $\epsilon$, $\frac{d\theta^{\prime}}{d\theta}=1+O_{C^{r-1}}(\epsilon)$,
so the parametric curve $(\theta^{\prime}(\theta;E),I^{\prime}(\theta;E))$ can
be represented as the graph
$(\theta^{\prime},I_{tan}^{\epsilon}(\theta^{\prime};E))$ where:

$\begin{split}I_{tan}^{\epsilon}(\theta^{\prime};E)&=I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta(\theta^{\prime},E))+\epsilon
f^{-}(I_{tan}(E),\theta(\theta^{\prime},E);\epsilon)+O(\epsilon^{2}),\\\
\theta&(\theta^{\prime},E)=\theta^{\prime}-\frac{1}{2}\cdot
T_{1}(E,I_{tan}(E))\cdot\omega_{2}(I_{tan}(E)).\end{split}$ (25)

∎

Recall that by lemma 3.3, on $\Sigma^{*}_{E}$, the $(q_{2},p_{2})$ values
inside $\sigma_{wall}^{\epsilon}(E)$ represent impacting trajectories which
reach the wall and are on $\Sigma_{E}^{w}$, whereas $(q_{2},p_{2})$ values
outside the iso-energy tangency curve represent non-impacting trajectories
crossing $\Sigma_{E}^{>}$ at $q_{1}$ value which is greater than $q_{1}^{w}$.
The tangency is the singular curve dividing between the two different types of
trajectories. Using the unperturbed action-angle coordinates to parametrize
$\Sigma^{*}_{E}$, by (18), for impacting trajectories

$H_{2}(I)+\epsilon V_{c}(q_{1}^{w},q_{2}(I,\theta))\leq H-V_{1}(q_{1}^{w})$
(26)

Hence on $\Sigma^{*}_{E}$ non-impacting trajectories are given by $(I,\theta)$
such that $I>I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)$ and impacting
trajectories by $I<I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta)$. Since the
backward evolution of the tangency curve $\sigma_{wall}^{\epsilon}(E)$ under
the backwards perturbed flow from $\Sigma_{1}$ to $\Sigma$ is the singularity
curve $\sigma^{\epsilon}(E)$ we conclude:

######  Lemma 3.6.

The singularity curve separates between impacting and non-impacting initial
conditions on $\Sigma_{E}$.

###### Proof.

Consider separately the backward evolution of iso-energy trajectories from
$\Sigma_{E}^{w}$ and from $\Sigma_{E}^{>}$ to $\Sigma_{E}$. The backward
Hamiltonian flow is orientation preserving, hence the backward image of
trajectories in $\Sigma_{E}^{w}$ (respectively $\Sigma_{E}^{>}$) are mapped to
the interior (respectively exterior) of the pre-image of their boundary, i.e.
the tangency curve. ∎

Let $\rho(z)=I_{tan}^{\epsilon}(\theta)-I$ denote the action-signed-distance
of an i.c. $z^{\theta,\rho,\epsilon}\in\Sigma_{E}$ from the singularity curve
at the angle $\theta$. By Lemma 3.5, since $I_{tan}^{\epsilon}(\theta)$ is
smooth, $\rho(z)=\rho(\theta,E)$ is well defined and is $C^{r}$. Hereafter, we
parameterize $z^{\theta,\rho,\epsilon}\in\Sigma_{E}$ by $(\theta,\rho)$:

$z^{\theta,\rho,\epsilon}:=(q_{1}=q_{1}^{\epsilon}(I_{tan}^{\epsilon}(\theta)-\theta,\rho,E),p_{1}=0,I=I_{tan}^{\epsilon}(\theta)-\theta,\rho),$
(27)

so $H(z^{\theta,\rho,\epsilon})=E$ and
$q_{1}^{\epsilon}(I,\theta,E)=q_{1,max}(E-H_{2}(I))+O(\epsilon)$ is found by
the IFT from Eq. (1). By definition, $\Pi
z^{\theta,0,\epsilon}=(I_{tan}^{\epsilon}(\theta),\theta)\in\sigma^{\epsilon}(E)$
and the unperturbed tangent trajectory is
$z_{tan}(t,\theta;E)=\Phi_{t}^{0}z^{\theta,0,0}$ . By Lemma 3.6 , for all
$\epsilon\in[0,\epsilon_{1}]$, $\rho>0$ corresponds to impacting i.c. of
$\mathcal{F}_{\epsilon}$ and $\rho\leqslant 0$ correspond to non-impacting
i.c. of this map. Notice that
$z^{\theta,\rho,0}=(q_{1,max}(E-H_{2}(I)),0,I_{tan}(E)-\theta,\rho)$, so, by
lemma 3.5,
$z^{\theta,\rho,\epsilon}=z^{\theta,\rho,0}+\mathcal{O}_{C^{r}}(\epsilon)$.

###  3.3 The image of the singularity curve

######  Proposition 3.7.

For sufficiently small $\epsilon$, the return map to $\Sigma_{E}$ of an
initial condition on the singularity curve,
$(\theta,I^{\epsilon}_{tan}(\theta))$, is of the form:

$\begin{split}\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan})&=I_{tan}^{\epsilon}(\theta)+\epsilon
f(\theta;\epsilon)\equiv\mathfrak{f}_{\epsilon}(\theta)\\\
\bar{\theta}_{tan}&=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}(I_{tan}(E);H)+\epsilon
g(I_{tan}^{\epsilon}(\theta),\theta;\epsilon)\end{split}$ (28)

where $\bar{I}_{tan}^{\epsilon}(\cdot)$ is the forward image of the tangency
curve on $\Sigma_{E}$, $\bar{\theta}_{tan}$ is the forward image of $\theta$,
$f,g$ are $C^{r}$ smooth in $\theta,\epsilon$ and

$f(\theta;\epsilon)=\int_{0}^{{T}_{1}^{\epsilon}(z^{\theta,0,\epsilon})}\\{I,V_{c}\\}\mid_{\Phi_{t}^{\epsilon}z^{\theta,0,\epsilon}}dt.$
(29)

###### Proof.

Let
$z^{\theta,0,\epsilon}=(q_{10},p_{10}=0,q_{20},p_{20})\in\Xi^{\epsilon}(E)\subset\Sigma_{E}$,
where $S_{2}(q_{20},p_{20})=(\theta,I_{tan}^{\epsilon}(\theta))$. As a tangent
trajectory is not affected by the impact, integration along the smooth
perturbed vector field yields:

$\begin{cases}\bar{I}_{tan}=I+\int_{0}^{{T}_{1}^{\epsilon}(z^{\theta,0,\epsilon})}\\{I,H_{int}+\epsilon
V_{c}\\}\mid_{\Phi_{t}^{\epsilon}z^{\theta,0,\epsilon}}dt=I_{tan}^{\epsilon}(\theta)+\epsilon
f(\theta;\epsilon)\\\
\bar{\theta}_{tan}=\theta+\int_{0}^{{T}_{1}^{\epsilon}(z^{\theta,0,\epsilon})}\\{\theta,H_{int}+\epsilon
V_{c}\\}\mid_{\Phi_{t}^{\epsilon}z^{\theta,0,\epsilon}}dt=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}+\epsilon g(\theta,I_{tan}^{\epsilon}(\theta);\epsilon)\end{cases}$
(30)

where $f,g$ are $C^{r}$ smooth, $2\pi$-periodic functions of $\theta$ (see
section 4 for the leading order calculation of $f$).
${T}_{1}^{\epsilon}(z^{\theta,0,\epsilon})$ denotes the return time to
$\Sigma_{E}$ of the perturbed tangent trajectory and
$T_{1}^{tan}={T}_{1}^{0}(z^{\theta,0,0})=T_{1}(I_{tan}(E);E)$ is the return
time of the unperturbed tangent trajectory. By definition,
$\bar{I}_{tan}=\mathfrak{f}_{\epsilon}(\theta)\equiv
I_{tan}^{\epsilon}(\theta)+\epsilon
f(I_{tan}^{\epsilon}(\theta),\theta;\epsilon)$, lies on the forward image of
the tangency curve at the angle $\bar{\theta}_{tan}$, and by Eq. (30)
$\frac{d\bar{\theta}_{tan}}{d\theta}=1+O_{C^{r}}(\epsilon)$, hence
$\bar{I}_{tan}=\mathfrak{f}_{\epsilon}(\theta)=\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan})$.
∎

###  3.4 A time-reversal symmetry of the singularity curve

The singularity line and its image are related by a symmetry (see [15] for a
similar symmetry observed in a 1 d.o.f harmonic oscillator impacting with a
harmonic oscillating wall):

######  Lemma 3.8.

The image of the singularity curve $\mathcal{\sigma}^{\epsilon}$ under
$\mathcal{F}_{\epsilon}$ is identical to its reflection about the $q_{2}$
axis, namely
$\bar{\sigma}^{\epsilon}:=\mathcal{F}_{\epsilon}\mathcal{\sigma}^{\epsilon}=\\{(\theta,I_{tan}^{\epsilon}(-\theta)),\theta\in[-\pi,\pi]\\}$.

###### Proof.

This is a result of a pointwise symmetry on the singularity curve which
follows from the mechanical form of the Hamiltonian. Let
$z_{0}=(q_{10},p_{10}=0,q_{20},p_{20})\in\Xi^{\epsilon}(E)\subset\Sigma_{E},$
let $\mathcal{R}_{2}z_{i}=(q_{1i},p_{1i}=0,q_{2i},-p_{2i})$ denote a
reflection about the $q_{2}$ axis and let $S_{2}(q_{2i},p_{2i})=S_{2}\Pi
z_{i}=(I_{i},\theta_{i})$. Since we consider i.c. on the singularity curve, by
definition, if $(I_{0},\theta_{0})=S_{2}(q_{20},p_{20})$ then
$I_{0}=I_{tan}^{\epsilon}(\theta_{0})$. Let
$z^{w}=(q_{1}^{w},0,q_{2}^{w},p_{2}^{w})=\Phi^{\epsilon}_{t^{-,\epsilon}(z_{0})}z_{0}$
and
$z_{1}=(q_{11},p_{11}=0,q_{21},p_{21})=\Phi^{\epsilon}_{t^{+,\epsilon}(z_{0})}z^{w}=\Phi^{\epsilon}_{t^{\epsilon}(z_{0})}z_{0}$
denote the image of $z_{0}$ at the star section and under the return map
respectively, so
$(q_{21},p_{21})=\mathcal{F}_{\epsilon}(I_{tan}^{\epsilon}(\theta_{0}),\theta_{0})$.
Then, from time reversal symmetry:

$\begin{split}\Phi^{\epsilon}_{t^{+,\epsilon}(z_{0})}(q_{1}^{w},0,q_{2}^{w},p_{2}^{w})&=(q_{11},p_{11}=0,q_{21},p_{21})\\\
\Rightarrow\Phi^{\epsilon}_{-t^{+,\epsilon}(z_{0})}(q_{1}^{w},0,q_{2}^{w},-p_{2}^{w})&=(q_{11},-p_{11}=0,q_{21},-p_{21})=\mathcal{R}_{2}z_{1}\\\
\Rightarrow\Phi^{\epsilon}_{t^{+,\epsilon}(z_{0})}\mathcal{R}_{2}z_{1}&=(q_{1}^{w},0,q_{2}^{w},-p_{2}^{w})\\\
\end{split}$ (31)

Thus, $\mathcal{R}_{2}z_{1}$ also belongs to the singularity curve and
$t^{-,\epsilon}(\mathcal{R}_{2}z_{1})=t^{+,\epsilon}(z_{0})$. Similarly,

$\begin{split}\Phi^{\epsilon}_{-t^{-,\epsilon}(z_{0})}(q_{1}^{w},0,q_{2}^{w},p_{2}^{w})&=(q_{10},p_{10}=0,q_{20},p_{20})\\\
\Rightarrow\Phi^{\epsilon}_{t^{-,\epsilon}(z_{0})}(q_{1}^{w},0,q_{2}^{w},-p_{2}^{w})&=(q_{10},-p_{10}=0,q_{20},-p_{20})=\mathcal{R}_{2}z_{0}\end{split}$
(32)

Thus

$\Phi^{\epsilon}_{{t^{-,\epsilon}(z_{0})+t^{+,\epsilon}(z_{0})}}\mathcal{R}_{2}z_{1}=\mathcal{R}_{2}z_{0}$
(33)

We deduce that $\Pi\mathcal{R}_{2}z_{1}\in\sigma^{\epsilon}(E)$ and that
$\mathcal{F}_{\epsilon}\Pi\mathcal{R}_{2}z_{1}=\Pi\mathcal{R}_{2}z_{0}\in\bar{\sigma}^{\epsilon}(E)$.
Since by the definition of action-angle coordinates for mechanical Hamiltonian
(see appendix)
$S_{2}(\Pi\mathcal{R}_{2}z_{i})=S_{2}(q_{2i},-p_{2i})=(I_{i},-\theta_{i})$ we
conclude that $\Pi\mathcal{R}_{2}z_{1}\in\sigma^{\epsilon}(E)$ implies that
$I_{1}=\bar{I}_{tan}^{\epsilon}(\theta_{1})=I_{tan}^{\epsilon}(-\theta_{1})$,
and, equivalently, $\Pi\mathcal{R}_{2}z_{0}\in\bar{\sigma}^{\epsilon}(E)$
implies that
$I_{tan}^{\epsilon}(\theta_{0})=\bar{I}_{tan}^{\epsilon}(-\theta_{0})$, as
claimed, see Figure 2.

∎

Figure 2: Two symmetric tangent trajectories projected to the $(q_{1},p_{1})$
(left) and $(q_{2},p_{2})$ (right) planes. In the $(q_{1},p_{1})$ plane the
starting point (solid circles) of a trajectory on $\Sigma_{E}$ is the ending
point (diamond) of the other, whereas in the $(q_{2},p_{2})$ plane the
starting and ending points are related by the time-reversal symmetry in
$\theta$. Here $q_{1}^{w}=-0.8$,
$V_{1}(q_{1})=2q_{1}^{2}+q_{1}^{3}+\frac{1}{4}q_{1}^{4}$,
$V_{2}(q_{2})=\frac{3}{2}q_{2}^{2},H=2$, $V_{c}(q_{1},q_{2})=q_{1}\cdot q_{2}$
and $\epsilon=0.3$.

Combining this symmetry and lemma 3.5 we conclude:

######  Corollary 3.9.

For sufficiently small $\epsilon$, the forward image of the singularity curve
can be represented as a graph in the unperturbed action-angle coordinates:
$\bar{\sigma}^{\epsilon}(E)=\\{(\theta,\bar{I}_{tan}^{\epsilon}(\theta)),\theta\in[-\pi,\pi]\\}$.

###  3.5 Proof of Theorem 1

Combining the above lemmas we establish Theorem 1: by lemma 3.6, for
$E>V_{1}(q_{1}^{w})$ and sufficiently small $\epsilon$, the singularity curve
$\sigma^{\epsilon}(E)$ is an isolated circle, dividing the cylinder
$\Sigma_{E}$ to impacting (below the curve) and non-impacting (above the
curve) regions. By lemma 3.5 for sufficiently small $\epsilon$, this circle is
a graph:
$\sigma^{\epsilon}(E)=\\{(\theta,I_{tan}^{\epsilon}(\theta)),\theta\in[-\pi,\pi]\\}$,
by corollary 3.9 its image under the map is also a graph,
$\bar{\sigma}^{\epsilon}(E)=\mathcal{F}_{\epsilon}\sigma^{\epsilon}=\\{(\theta,\bar{I}_{tan}^{\epsilon}(\theta)=I_{tan}^{\epsilon}(2\pi-\theta)),\theta\in[-\pi,\pi]\\}$,
and by lemma 3.2, 3.5 and corollary 3.9 both $I_{tan}^{\epsilon}(\theta)$ and
$\bar{I}_{tan}^{\epsilon}(\theta)$ are $C^{r}$ $\epsilon-$close to the
constant function $I_{tan}(E)$ and
$I_{tan}^{\epsilon}(\theta)=\bar{I}_{tan}^{\epsilon}(-\theta)$.

##  4 The near tangent return map

Near tangency, the unperturbed map $\mathcal{F}_{0}$ is symplectic and $C^{0}$
but is not smooth due to the non-smooth dependence of the rotation number on
$I$, see Eqs. (5-7). This singularity leads to non-trivial behavior under
perturbation. We first construct the perturbed near tangent return map by
separating the approximations of the perturbed near tangent trajectories to
impacting and non-impacting, according to their position with respect to the
singularity curve on $\Sigma_{E}$. We then construct a change of coordinates
bringing the near tangent map to the form (9).

###  4.1 The perturbed near tangent return map

######  Proposition 4.1.

For sufficiently small $\epsilon$, the near-tangent local return map to the
cross-section $\Sigma_{E}$,
$\mathcal{F}_{\epsilon}^{loc}:(\theta,I)\rightarrow(\bar{\theta},\bar{I})$ is
of the form :

$\displaystyle I\geq I^{\epsilon}_{tan}(\theta):$
$\displaystyle\begin{cases}\bar{I}=I+\bar{I}^{\epsilon}_{tan}(\bar{\theta})-I^{\epsilon}_{tan}(\theta)+\mathcal{O}_{C^{r-1}}(\epsilon(I-I^{\epsilon}_{tan}(\theta)))\\\
\begin{aligned} \bar{\theta}=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}+&\tau(I_{tan}(E))\cdot(I-I^{\epsilon}_{tan}(\theta))+\mathcal{O}_{C^{r-1}}(\epsilon,(I-I^{\epsilon}_{tan}(\theta))^{2})\end{aligned}\end{cases}$
(34a) $\displaystyle I<I^{\epsilon}_{tan}(\theta):$
$\displaystyle\begin{cases}\bar{I}=I+\bar{I}^{\epsilon}_{tan}(\bar{\theta})-I^{\epsilon}_{tan}(\theta)+\mathcal{O}_{C^{r-2}}(\epsilon\sqrt{I^{\epsilon}_{tan}(\theta)-I}\
)\\\ \begin{aligned} \bar{\theta}=\theta+\omega_{2}&(I_{tan}(E))\cdot
T_{1}^{tan}(E)+G_{s}(I-I^{\epsilon}_{tan}(\theta))+\mathcal{O}_{C^{r-2}}(\epsilon,(I-I^{\epsilon}_{tan}(\theta))^{2},\epsilon\sqrt{I^{\epsilon}_{tan}(\theta)-I})\end{aligned}\end{cases}$
(34b)

where $\bar{I}^{\epsilon}_{tan}(\bar{\theta})$ is the image of
$I_{tan}^{\epsilon}(\theta)$ under the map parametrized by $\bar{\theta}$ (see
Proposition 3.7),
$\bar{I}^{\epsilon}_{tan}(\bar{\theta})-I^{\epsilon}_{tan}(\theta)=\epsilon\int_{0}^{T^{tan}_{1}(E)}\\{I,V_{c}\\}\mid_{z_{tan}(t,\theta;E)}dt+\mathcal{O}_{C^{r}}(\epsilon^{2})$
and $G_{s}(\cdot)$ is the function of Eq. (10) (continuous with a square root
singularity at the origin for negative arguments).

Figure 3: Iso-energy impacting, tangent and non-impacting trajectories
segments in-between crossings of the return section $\Sigma_{E}$. (a)
Projection to the $(q_{1},p_{1})$ plane. (b) Projection to the $(q_{2},p_{2})$
plane. Here, all segments meet $\Sigma_{E}^{*}$ with the same angle
$\theta^{w}$ (open circles in (b)), yet, each initial condition (filled
circles), $z^{\theta,\rho,\epsilon}$, has a different phase $\theta$ and a
corresponding distinct action distance $\rho$. Parameters are as in Figure 2.

###### Proof.

Let $\epsilon_{1}>0$ be sufficiently small so that Theorem 1 holds, and
consider $\epsilon\in[0,\epsilon_{1}]$. Then, i.c. on $\Sigma_{E}$ can be
parameterized by $\theta,\rho$ (recall Eq. (27)) and:

$\mathcal{F}_{\epsilon}^{loc}\Pi
z^{\theta,\rho,\epsilon}=\Pi\Phi^{\epsilon,im}_{t^{\epsilon}(z)}z,\quad\Phi^{\epsilon,im}_{t^{\epsilon}(z)}z=\begin{cases}\Phi^{\epsilon}_{t^{\epsilon}(z)}z&\text{no
impact, }\rho\leqslant 0\\\
\Phi^{\epsilon}_{t^{+,\epsilon}(z)}\circ\mathcal{R}_{1}\circ\Phi^{\epsilon}_{t^{-,\epsilon}(z)}z&\text{with
impact, }\rho>0.\\\ \end{cases}$ (35)

where, for shorthand notation, we omit the labeling of $z$ when it is
inessential. By Theorem 1, the division to these two cases occurs exactly at
the singularity curve $\sigma^{\epsilon}(E)$, and by lemma 3.4,
$t^{-,\epsilon}(z)$, the crossing time of $\Sigma_{E}^{*}$, and
$t^{+,\epsilon}(z)$, the travel time from $\Sigma_{E}^{*}$ to $\Sigma_{E}$ are
uniquely defined by $z$ and limit to $\frac{1}{2}T_{1}(I_{tan}(E))$ as
$\epsilon\rightarrow 0$, thus they are finite. Next, we calculate the
integrals of $\\{I,H\\}$ and $\\{\theta,H\\}$ along the perturbed trajectory
segments on the finite time intervals $[0,t^{-,\epsilon}(z))$ and
$(t^{-,\epsilon}(z),\tilde{T}_{1}^{\epsilon}(z)=t^{-,\epsilon}(z)+t^{+,\epsilon}(z)]$,
approximate these integrals along some properly defined unperturbed
approximations (see below and Figure 1), and express the results in terms of
the image of the singularity curve.

Denote by $z_{sm}^{\theta,\rho,\epsilon}(t)$ the trajectory of the smooth
Hamiltonian system ($b=0$) emanating from $z^{\theta,\rho,\epsilon}$ (so
$z^{\theta,\rho,\epsilon}_{sm}(0)=z^{\theta,\rho,\epsilon},z_{sm}^{\theta,\rho,\epsilon}(t)=\Phi_{t}^{\epsilon}z^{\theta,\rho,\epsilon}$).
This solution depends smoothly on i.c. and on $\epsilon$, so, for small
$\epsilon,\rho$ and for all $0\leq t\leq
T_{1}^{tan}(E)+\eta,\eta\in[0,\frac{1}{4}T_{1}^{tan}]$ :

$z_{sm}^{\theta,\rho,\epsilon}(t)=z_{sm}^{\theta,\rho,0}(t)+\mathcal{O}_{C^{r}}(\epsilon)=z_{sm}^{\theta,0,0}(t)+\mathcal{O}_{C^{r}}(\epsilon,\rho)=z_{tan}(t,\theta;E)+\mathcal{O}_{C^{r}}(\epsilon,\rho).$
(36)

Let
${T}_{1}^{\theta,\rho,\epsilon}={T}_{1}^{\epsilon}(z^{\theta,\rho,\epsilon})$
denote the return time of $z_{sm}^{\theta,\rho,\epsilon}(t)$ to the cross-
section $\Sigma_{E}$, $t^{h,\epsilon}(z^{\theta,\rho,\epsilon})$ the travel
time of $z_{sm}^{\theta,\rho,\epsilon}(t)$ from $\Sigma_{E}$ to the mid
section $\Sigma_{E}^{<}\cup\Sigma_{E}^{>}$ and by
$t^{\star}(z^{\theta,\rho,\epsilon})$ its travel time from
$z^{w}(z^{\theta,\rho,\epsilon})=z_{sm}^{\theta,\rho,\epsilon}(t^{-,\epsilon}(z^{\theta,\rho,\epsilon}))\in\Sigma^{w}_{E}$
to $\Sigma_{E}^{<}$, so
$t^{\star}=t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{-,\epsilon}(z^{\theta,\rho,\epsilon})$
and $t^{\star}(z^{\theta,\rho,0})=\frac{1}{2}\Delta
t_{travel}(J(I_{tan}(E)-\rho,E))$. By smooth dependence on i.c. and parameters
(Eq. (36)):

$T_{1}^{\theta,\rho,\epsilon}=T_{1}^{\theta,\rho,0}+\mathcal{O}_{C^{r}}(\epsilon)=T_{1}^{tan}(E)-\rho\cdot\frac{d}{dI}T_{1}(J(I))|_{I_{tan}(E)}+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2})$
(37)

and

$t^{h,\epsilon}(z^{\theta,\rho,\epsilon})=\frac{1}{2}T_{1}^{\theta,\rho,0}+\mathcal{O}_{C^{r}}(\epsilon)=\frac{1}{2}T_{1}^{tan}(E)-\frac{1}{2}\rho\cdot\frac{d}{dI}T_{1}(J(I))|_{I_{tan}(E)}+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2}).$
(38)

In particular,
$T_{1}^{\theta,0,\epsilon}=T_{1}^{\epsilon}(I^{\epsilon}_{tan}(\theta),\theta;E)$
is the return time to $\Sigma_{E}$ of the perturbed tangent trajectory
$z_{sm}^{\theta,0,\epsilon}(t)$, and
$T_{1}^{\theta,0,0}=T_{1}^{tan}(E)=T_{1}(J(I_{tan}(E),E))$, the return time of
unperturbed tangent trajectory $z_{sm}^{\theta,0,0}(t)$ as well
$T_{1}^{\theta,\rho,0}=T_{1}^{\rho}=T_{1}(J(I_{tan}(E)-\rho,E))$, the return
time of unperturbed smooth trajectory with actions
$(J,I)=(J(I_{tan}(E)-\rho,E),I_{tan}(E)-\rho)$ are independent of $\theta$.

##### The return map for non-impacting i.c., Eq. 34a:

For non-positive $\rho$ (green and blue trajectory segments in Figures 1 and
3), the trajectory segment between returns does not impact, so
$z_{im}^{\theta,\rho,\epsilon}(t)|_{\rho\leqslant
0}=z_{sm}^{\theta,\rho,\epsilon}(t)$, and smooth standard perturbation theory
applies:

$\begin{split}\bar{I}=I(\mathcal{F}_{\epsilon}\Pi
z^{\theta,\rho,\epsilon})&=I(\Pi
z^{\theta,\rho,\epsilon})+\int_{0}^{{T}_{1}^{\theta,\rho,\epsilon}}\\{I,H_{int}+\epsilon
V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&=I_{tan}^{\epsilon}(\theta)-\rho+\epsilon\int_{0}^{T_{1}^{\theta,0,\epsilon}+\mathcal{O}_{C^{r}}(\rho)}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,\epsilon}(t)+\mathcal{O}_{C^{r}}(\rho)}dt\\\
&=\mathfrak{f}_{\epsilon}(\theta)-\rho+\mathcal{O}_{C^{r}}(\epsilon\rho)\\\
&=I_{tan}^{\epsilon}(\theta)-\rho+\epsilon\int_{0}^{T_{1}^{\theta,0,0}+\mathcal{O}_{C^{r}}(\epsilon)}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,0}(t)+\mathcal{O}_{C^{r}}(\rho,\epsilon)}dt+\mathcal{O}_{C^{r}}(\epsilon\rho)\\\
&=I_{tan}^{\epsilon}(\theta)-\rho+\epsilon\int_{0}^{T_{1}^{tan}(E)}\\{I,V_{c}\\}\mid_{z_{tan}(t,\theta;E)(t)}dt+\mathcal{O}_{C^{r}}(\epsilon\rho,\epsilon^{2})\end{split}$
(39)

where, as in Proposition 3.7 (i.e. Eq. (30)),

$\begin{split}\mathfrak{f}_{\epsilon}(\theta):=I(\mathcal{F}_{\epsilon}\Pi
z^{\theta,0,\epsilon})=\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan})&=I_{tan}^{\epsilon}(\theta)+\epsilon\int_{0}^{{T}_{1}^{\theta,0,\epsilon}}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,\epsilon}(t)}dt\\\
&=I_{tan}^{\epsilon}(\theta)+\epsilon
f(I_{tan}^{\epsilon}(\theta),\theta;\epsilon)\\\
&=I_{tan}^{\epsilon}(\theta)+\epsilon\int_{0}^{T^{tan}_{1}(E)}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,0}(t)}dt+\mathcal{O}_{C^{r}}(\epsilon^{2})\end{split}$
(40)

is the action of the image of $z^{\theta,0,\epsilon}\in\sigma^{\epsilon}$,
namely, $\mathcal{F}_{\epsilon}\Pi
z^{\theta,0,\epsilon}=(\mathfrak{f}_{\epsilon}(\theta)=\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan}),\bar{\theta}_{tan})\in\bar{\sigma}^{\epsilon}$.
In particular, the leading order term of the action of the singularity curve
image is:

$f(I_{tan}^{\epsilon}(\theta),\theta;0)=\int_{0}^{T^{tan}_{1}(E)}\\{I,V_{c}\\}\mid_{z_{tan}(t,\theta;E)}dt.$
(41)

The leading order dependence of $\bar{\theta}$ on $\rho$ is similarly found:

$\begin{split}\bar{\theta}=\theta(\mathcal{F}_{\epsilon}\Pi
z^{\theta,\rho,\epsilon})&=\theta+\int_{0}^{T_{1}^{\theta,\rho,\epsilon}}\\{\theta,H_{int}+\epsilon
V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&=\theta+\int_{0}^{T_{1}^{\theta,\rho,0}+\mathcal{O}_{C^{r}}(\epsilon)}(\\{\theta,H_{int}\\}+\epsilon\\{\theta,V_{c}\\})\mid_{z_{sm}^{\theta,\rho,0}(t)+\mathcal{O}_{C^{r}}(\epsilon)}dt\\\
&=\theta+\omega_{2}(I_{tan}(E)-\rho)\cdot
T_{1}^{\rho}+\mathcal{O}_{C^{r}}(\epsilon)\\\
&=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}^{\rho}-\rho\cdot\omega_{2}^{\prime}(I_{tan}(E))\cdot
T_{1}^{\rho}+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2})\\\
&=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}(E)-\rho\cdot\tau(I_{tan}(E))+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2})\end{split}$
(42)

where $\tau(I)$ is the twist of the iso-energy return map of the smooth
Hamiltonian system (see Eq. (11)), so the second line of Eq. 34a is
established.

Finally, we show that Eq. (39) implies the first line of Eq. 34a. By
Proposition 3.7, on the singularity curve:

$\begin{split}\bar{\theta}_{tan}=\bar{\theta}(z^{\theta,0,\epsilon})=\theta(\mathcal{F}_{\epsilon}\Pi
z^{\theta,0,\epsilon})&=\theta+\int_{0}^{T_{1}^{\theta,0,\epsilon}}\omega_{2}(I(z_{sm}^{\theta,0,\epsilon}(t))+\epsilon\\{\theta,V_{c}\\}\mid_{z_{sm}^{\theta,0,\epsilon}(t)}dt\\\
&=\theta+\omega_{2}(I_{tan}(E))\cdot T_{1}^{tan}(E)+\epsilon
g(I_{tan}^{\epsilon}(\theta),\theta;\epsilon),\end{split}$ (43)

where the $C^{r}$ smooth function
$g(I_{tan}^{\epsilon}(\theta),\theta;\epsilon)$ includes integration along the
perturbed tangent segment (and does not have a simple approximation by the
unperturbed dynamics). For small negative $\rho$ we obtain (see first line of
Eq. (42)) that

$\bar{\theta}(z^{\theta,\rho,\epsilon})=\theta+\int_{0}^{T_{1}^{\theta,0,\epsilon}+\mathcal{O}_{C^{r}}(\rho)}\\{\theta,H_{int}+\epsilon
V_{c}\\}\mid_{z_{sm}^{\theta,0,\epsilon}(t)+\mathcal{O}_{C^{r}}(\rho)}dt=\bar{\theta}_{tan}+\mathcal{O}_{C^{r}}(\rho),$

so, as $\bar{I}_{tan}^{0}(\cdot)=I_{tan}(E)$ and is independent of its
argument,
$\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan})=\bar{I}_{tan}^{\epsilon}(\bar{\theta})+\mathcal{O}_{C^{r-1}}(\epsilon\rho)$,
and thus, by Eq. (39):

$\bar{I}(z^{\theta,\rho,\epsilon})=\bar{I}_{tan}^{\epsilon}(\bar{\theta})-\rho+\mathcal{O}_{C^{r-1}}(\epsilon\rho)=I+\bar{I}^{\epsilon}_{tan}(\bar{\theta})-I^{\epsilon}_{tan}(\theta)+\mathcal{O}_{C^{r-1}}(\epsilon(I_{tan}^{\epsilon}(\theta)-I))$
(44)

thus Eq. 34a is established.

##### The return map for impacting i.c. (Eq. 34b):

For negative $\rho$ (the red trajectory segment in Figures 1 and 3), we divide
$z_{im}^{\theta,\rho,\epsilon}(t)$ to two smooth segments, before and after
the impact. We employ regular perturbation theory for estimating the change in
$(\theta,I)$ along these segments (see also [23]), showing that these depend
smoothly on the travel times to and from the impact. We then study the
singular dependence of the travel times and establish (34b). To this aim we
introduce the following notations:

  * •

Let
$z_{im}^{\theta,\rho,\epsilon}(t):=\Phi^{\epsilon,im}_{t}z^{\theta,\rho,\epsilon}$
denote the wall trajectory segment with initial condition
$z^{\theta,\rho,\epsilon}_{im}(0)=z^{\theta,\rho,\epsilon}$ and
$t\in[0,\tilde{T}_{1}^{\theta,\rho,\epsilon}]$, where
$\tilde{T}_{1}^{\theta,\rho,\epsilon}=t^{\epsilon}(z^{\theta,\rho,\epsilon})$
is the return time of the wall trajectory $z_{im}^{\theta,\rho,\epsilon}(t)$
to the cross-section $\Sigma_{E}$ and
$t^{-,\epsilon}(z^{\theta,\rho,\epsilon}),t^{+,\epsilon}(z^{\theta,\rho,\epsilon})$
denote the corresponding travel times between $\Sigma_{E}$ to $\Sigma^{*}_{E}$
and back. Thus, $\mathcal{F}_{\epsilon}^{loc}\Pi z^{\theta,\rho,\epsilon}=\Pi
z_{im}^{\theta,\rho,\epsilon}(\tilde{T}_{1}^{\theta,\rho,\epsilon})$.

  * •

Let
$z^{w}=\Phi^{\epsilon}_{t^{-,\epsilon}(z)}z^{\theta,\rho,\epsilon}=(q_{1}^{w},p_{1}^{w},\theta^{w},I^{w})$
and let $\rho^{w}$ denote the action-distance from tangency at the wall
${}\rho^{w}=I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})-I^{w}$, where
$I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})$ is the tangency circle at the
wall.

  * •

Let
$\hat{z}_{sm}^{\theta,\rho,\epsilon}(t):=\Phi^{\epsilon}_{t-\hat{T}_{1}^{\theta,\rho,\epsilon}}\circ\Phi^{\epsilon}_{t^{+,\epsilon}(z)}\circ\mathcal{R}_{1}\circ\Phi^{\epsilon}_{t^{-,\epsilon}(z)}z^{\theta,\rho,\epsilon}$
denote the smooth flow from the backward smooth pre-image of
$z_{im}^{\theta,\rho,\epsilon}(\tilde{T}_{1}^{\theta,\rho,\epsilon})=\hat{z}_{sm}^{\theta,\rho,\epsilon}(\hat{T}_{1}^{\theta,\rho,\epsilon})$,
so
$\hat{T}_{1}^{\theta,\rho,\epsilon}=T_{1}^{\epsilon}(\hat{z}_{sm}^{\theta,\rho,\epsilon}(0))$.
Equivalently, there exists
$\hat{\rho}=\hat{\rho}(\theta,\rho,\epsilon),\hat{\theta}=\hat{\theta}(\theta,\rho,\epsilon)$
s.t
$\hat{z}_{sm}^{\theta,\rho,\epsilon}(t)=z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)$
and $\hat{I}=I_{tan}^{\epsilon}(\hat{\theta})-\hat{\rho}$ (the dependence of
$(\hat{\theta},\hat{\rho})$ is non-smooth near tangency, i.e. near $\rho=0$,
see below). By construction, this is the smooth trajectory segment which
coincides with the impacting trajectory after the impact:
$z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)=z_{im}^{\theta,\rho,\epsilon}(t),\,t\in[t^{+,\epsilon}(z^{\theta,\rho,\epsilon}),\tilde{T}_{1}^{\theta,\rho,\epsilon}]$.

  * •

Let $t^{\star}$ and $\hat{t}^{\star}$ denote the forward and backward flight
times from the wall section and its reflection to the auxiliary section
$\Sigma^{<}_{E}$, so $z_{min}:=\Phi^{\epsilon}_{t^{\star}}z^{w}$ and
$\hat{z}_{min}:=\Phi^{\epsilon}_{-\hat{t}^{\star}}\mathcal{R}_{1}z^{w}$ are in
$\Sigma^{<}_{E}$,
$t^{\star}(z^{\epsilon,\rho,\theta})=t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{-,\epsilon}(z^{\theta,\rho,\epsilon})$
and
$\hat{t}^{\star}(z^{\theta,\rho,\epsilon})=T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})-t^{+,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})$).

When $\epsilon=0$ the return times of the wall and smooth trajectories (i.e.
of $z_{im}^{\theta,\rho,0}(t)$ and $z_{sm}^{\theta,\rho,0}(t)$) do not depend
on $\theta$, so
$T_{1}^{\theta,\rho,0}=T_{1}^{\rho}=T_{1}(J(I_{tan}(E)-\rho,E))$, similarly
$\tilde{T}_{1}^{\theta,\rho,0}=\tilde{T}_{1}^{\rho}=\tilde{T}_{1}(J(I_{tan}(E)-\rho,E))$,
$\hat{\rho}(\theta,\rho,0)=\rho,\hat{\theta}(\theta,\rho,0)=\theta-2\Delta
t_{travel}(J(I_{tan}(E)-\rho,E))\omega_{2}(I_{tan}(E)-\rho))$, and
$t^{\star}(z^{\theta,\rho,0})=\hat{t}^{\star}(z^{\theta,\rho,0})=\Delta
t_{travel}(J(I_{tan}(E)-\rho,E))$.

Next, by integrating the brackets $\\{I,H\\},\\{\theta,H\\}$ along properly
chosen segments of $z_{sm}^{\theta,\rho,\epsilon}(t)$ and
$z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)$ we establish the leading order
behavior of the return map and show that the error terms are small and depend
smoothly on $t^{\star}$ and $\hat{t}^{\star}$. We then establish that these
flight times depend smoothly on $\sqrt{\rho^{w}}$. We complete the proof by
showing that functions that are small and depend smoothly on $\sqrt{\rho^{w}}$
also depend smoothly on $\sqrt{\rho}$, so that the error terms are indeed of
the required form.

######  Lemma 4.2.

The functions $\bar{I}=I(\mathcal{F}_{\epsilon}\Pi
z^{\theta,\rho,\epsilon}),\bar{\theta}=\theta(\mathcal{F}_{\epsilon}\Pi
z^{\theta,\rho,\epsilon})$ and
$I^{w}=I(\Phi^{\epsilon}_{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}z^{\theta,\rho,\epsilon}),$
$\theta^{w}=\theta(\Phi^{\epsilon}_{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}z^{\theta,\rho,\epsilon})$
are smooth functions of
$t^{\star},\hat{t}^{\star},\hat{\rho},\hat{\theta},\theta,\rho,\epsilon$
satisfying:

$\begin{split}\bar{I}&=I+(\bar{I}_{tan}^{\epsilon}(\bar{\theta})-I_{tan}^{\epsilon}(\theta))+\mathcal{O}_{C^{r}}(\rho\epsilon,\hat{\rho}\epsilon,t^{\star}\epsilon,\hat{t}^{\star}\epsilon,(\hat{\theta}-\theta)\epsilon,\rho^{2})\\\
\bar{\theta}&=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}(E)-\rho\cdot\tau(I_{tan}(E))-\omega_{2}(I_{tan}(E))\cdot(t^{\star}+\hat{t}^{\star})\\\
&\qquad\qquad+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2},\rho-\hat{\rho},(\rho-\hat{\rho})\hat{t}^{\star},\epsilon
t^{\star},\epsilon\hat{t}^{\star}).\end{split}$ (45)

and

$\begin{split}I^{w}&=I+\epsilon\int_{0}^{\frac{1}{2}T_{1}^{tan}}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,0}(t)}dt+\mathcal{O}_{C^{r}}(\rho\epsilon,t^{\star}\epsilon,\epsilon^{2})\\\
&=\hat{I}+\epsilon\int_{0}^{\frac{1}{2}T_{1}^{tan}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},0,0}(t)}dt+\mathcal{O}_{C^{r}}(\hat{\rho}\epsilon,\hat{t}^{\star}\epsilon,\epsilon^{2})\\\
\theta^{w}&=\theta+\frac{1}{2}\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}(E)-\frac{1}{2}\rho\cdot\tau(I_{tan}(E))-\omega_{2}(I_{tan}(E))\cdot
t^{\star}+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2},\epsilon
t^{\star})\end{split}$ (46)

###### Proof.

The leading order change in the action is calculated by integrating the
$\\{I,H\\}$ bracket along $z_{sm}^{\theta,\rho,\epsilon}(t)$ before the impact
and along
$\hat{z}_{sm}^{\theta,\rho,\epsilon}(t)=z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)$
after the impact, and approximating each segment separately by the
corresponding smooth tangent trajectory:

$\begin{split}\bar{I}=I(\mathcal{F}_{\epsilon}\Pi
z^{\theta,\rho,\epsilon})&=I(\Pi
z^{\theta,\rho,\epsilon})+\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})+t^{+,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,H_{int}+\epsilon
V_{c}\\}\mid_{z_{im}^{\theta,\rho,\epsilon}(t)}dt\\\
&=I+\epsilon\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt+\\\
&\qquad\qquad\
+\epsilon\int_{\hat{T}_{1}^{\theta,\rho,\epsilon}-t^{+,\epsilon}(z^{\theta,\rho,\epsilon})}^{\hat{T}_{1}^{\theta,\rho,\epsilon}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt\\\
&=I+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt-\epsilon\int_{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})+t^{\star}}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&\qquad\
+\epsilon\int_{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})}^{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt-\epsilon\int_{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})}^{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}^{\star}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}\\\
&\\\
&=I+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\theta,0,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,\epsilon}(t)}dt+\mathcal{O}_{C^{r}}(\rho\epsilon)-\epsilon\int_{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})+t^{\star}}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&\qquad\
+\epsilon\int_{T_{1}^{\hat{\theta},0,\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},0,\epsilon})}^{T_{1}^{\hat{\theta},0,\epsilon}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},0,\epsilon}(t)}dt+\mathcal{O}_{C^{r}}(\hat{\rho}\epsilon)-\epsilon\int_{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})}^{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}^{\star}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}\\\
&=I+(\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan})-I_{tan}^{\epsilon}(\theta))+\mathcal{O}_{C^{r}}(\rho\epsilon,\hat{\rho}\epsilon,t^{\star}\epsilon,\hat{t}^{\star}\epsilon,\epsilon(\hat{\theta}-\theta))\end{split}$
(47)

where, we used the smooth dependence of
$t^{h,\epsilon}(z^{\theta,\rho,\epsilon})$ on parameters and i.c., that all
the integrands are smooth and that the integration periods are finite. It
follows that the non-smooth behavior arises only from the singular dependence
of the lengths of the intervals of integration ($t^{\star}$ and
$\hat{t}^{\star}$) and from the non-smooth dependence of $\hat{\rho}$ and
$\hat{\theta}$ on $\theta,\rho$. So, up to the different form of the error
terms we have the same expression as in Eq. (39), and, to establish the
leading order behavior we again only need to show that
$\bar{I}_{tan}^{\epsilon}(\bar{\theta}_{tan}(\theta))$ and
$\bar{I}_{tan}^{\epsilon}(\bar{\theta})$ are close in the same sense. Since
$\bar{I}_{tan}^{\epsilon}$ is a smooth function and its dependence on its
argument is of order $\epsilon$, we establish this by showing that
$(\bar{\theta}-\bar{\theta}_{tan})=\mathcal{O}_{C^{r}}(\hat{\theta}-\theta,t^{\star},\hat{t}^{\star},\rho,\hat{\rho})$.
Indeed, the image of the angle is:

$\begin{split}\bar{\theta}(z^{\theta,\rho,\epsilon})&=\theta+\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}\\{\theta,H\\}\mid_{z_{im}^{\theta,\rho,\epsilon}(t)}dt+\int_{t^{+,\epsilon}(z^{\theta,\rho,\epsilon})}^{\tilde{T}_{1}^{\theta,\rho,\epsilon}}\\{\theta,H\\}\mid_{z_{im}^{\theta,\rho,\epsilon}(t)}dt\\\
&=\theta+\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}+\int_{t^{+,\epsilon}(z^{\theta,\rho,\epsilon})}^{\tilde{T}_{1}^{\theta,\rho,\epsilon}}[\omega_{2}(I(\cdot))+\epsilon\\{\theta,V_{c}\\}]\mid_{z_{im}^{\theta,\rho,\epsilon}(t)}dt\\\
&=\theta+\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}-\int_{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}[]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}\\\
&\qquad\
+\int_{T_{1}^{\epsilon,\hat{\rho},\hat{\theta}}-t^{h,\epsilon}(z^{\epsilon,\hat{\rho},\hat{\theta}})}^{T_{1}^{\epsilon,\hat{\rho},\hat{\theta}}}-\int^{T_{1}^{\epsilon,\hat{\rho},\hat{\theta}}-t^{+,\epsilon}(z^{\theta,\rho,\epsilon})}_{T_{1}^{\epsilon,\hat{\rho},\hat{\theta}}-t^{h,\epsilon}(z^{\epsilon,\hat{\rho},\hat{\theta}})}[]\mid_{z_{sm}^{\epsilon,\hat{\rho},\hat{\theta}}(t)}dt\end{split}$
(48)

Hence, it is close to the tangent angle image,
$\bar{\theta}_{tan}(\theta)=\bar{\theta}(z^{\theta,0,\epsilon})$:

$\begin{split}\bar{\theta}(z^{\theta,\rho,\epsilon})&=\theta+\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}[]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}+\int_{T_{1}^{\epsilon,\hat{\rho},\hat{\theta}}-t^{h,\epsilon}(z^{\epsilon,\hat{\rho},\hat{\theta}})}^{T_{1}^{\epsilon,\hat{\rho},\hat{\theta}}}[]\mid_{z_{sm}^{\epsilon,\hat{\rho},\hat{\theta}}(t)}dt+\mathcal{O}_{C^{r}}(t^{\star},\hat{t}^{\star})\\\
&=\theta+\int_{0}^{t^{h,\epsilon}(z^{\theta,0,\epsilon})}[]\mid_{z_{sm}^{\theta,0,\epsilon}(t)}+\int_{T_{1}^{\epsilon,0,\hat{\theta}}-t^{h,\epsilon}(z^{\epsilon,0,\hat{\theta}})}^{T_{1}^{\epsilon,0,\hat{\theta}}}[]\mid_{z_{sm}^{\epsilon,0,\hat{\theta}}(t)}dt+\mathcal{O}_{C^{r}}(t^{\star},\hat{t}^{\star},\rho,\hat{\rho})\\\
&=\bar{\theta}(z^{\theta,0,\epsilon})+\mathcal{O}_{C^{r}}(\hat{\theta}-\theta,t^{\star},\hat{t}^{\star},\rho,\hat{\rho}).\end{split}$
(49)

Thus, we established the claimed result for $\bar{I}$.

The form of $I^{w}$ follows by the same approximations as in (47):

$\begin{split}I^{w}(z^{\theta,\rho,\epsilon})&=I(\Pi
z^{\theta,\rho,\epsilon})+\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,H_{int}+\epsilon
V_{c}\\}\mid_{z_{im}^{\theta,\rho,\epsilon}(t)}dt\\\
&=I+\epsilon\int_{0}^{\frac{1}{2}T_{1}^{tan}}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,0}(t)}dt+\mathcal{O}_{C^{r}}(\rho\epsilon,t^{\star}\epsilon,\epsilon^{2}).\end{split}$
(50)

and similarly

$\begin{split}I^{w}(z^{\theta,\rho,\epsilon})&=I(\Pi
z^{\hat{\theta},\hat{\rho},\epsilon})+\int_{0}^{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{+,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,H_{int}+\epsilon
V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt\\\
&=\hat{I}+\int_{0}^{t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}^{\star}}\\{I,H_{int}+\epsilon
V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt\\\
&=\hat{I}+\epsilon\int_{0}^{\frac{1}{2}T_{1}^{tan}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},0,0}(t)}dt+\mathcal{O}_{C^{r}}(\hat{\rho}\epsilon,\hat{t}^{\star}\epsilon,\epsilon^{2}).\end{split}$
(51)

Finally, notice that to leading order in $\epsilon$, Eq. (48) becomes

$\begin{split}\bar{\theta}(z^{\theta,\rho,\epsilon})&=\theta+\int_{0}^{t^{h,0}(z^{\theta,\rho,0})}-\int_{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{\star}}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}\omega_{2}(I(\cdot))\mid_{z_{sm}^{\theta,\rho,0}(t)}dt+\mathcal{O}_{C^{r}}(\epsilon,\epsilon
t^{\star})\\\
&\qquad+\int_{T_{1}^{\hat{\theta},\hat{\rho},0}-t^{h,0}(z^{\hat{\theta},\hat{\rho},0})}^{T_{1}^{\hat{\theta},\hat{\rho},0}}-\int^{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}^{\star}}_{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})}\omega_{2}(I(\cdot))\mid_{z_{sm}^{\hat{\theta},\hat{\rho},0}(t)}+\mathcal{O}_{C^{r}}(\epsilon,\epsilon\hat{t}^{\star})\\\
&\\\
&=\theta+\omega_{2}(I_{tan}(E)-\rho)\cdot(\frac{1}{2}T_{1}^{\rho}-t^{\star})+\mathcal{O}_{C^{r}}(\epsilon,\epsilon
t^{\star})\\\
&\qquad\qquad+\omega_{2}(I_{tan}(E)-\hat{\rho})\cdot(\frac{1}{2}T_{1}^{\hat{\rho}}-\hat{t}^{\star})+\mathcal{O}_{C^{r}}(\epsilon,\epsilon\hat{t}^{\star})\\\
&=\theta+\omega_{2}(I_{tan}(E)-\rho)\cdot(T_{1}^{\rho}-(t^{\star}+\hat{t}^{\star}))+\mathcal{O}_{C^{r}}(\epsilon,\epsilon
t^{\star},\epsilon\hat{t}^{\star},\rho-\hat{\rho},(\rho-\hat{\rho})\hat{t}^{\star})\end{split}$
(52)

So we established (compare with Eq. (42)):

$\begin{split}\bar{\theta}(z^{\theta,\rho,\epsilon})&=\theta+\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}(E)-\rho\cdot\tau(I_{tan}(E))-\omega_{2}(I_{tan}(E))\cdot(t^{\star}+\hat{t}^{\star})\\\
&\qquad\qquad+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2},\rho-\hat{\rho},(\rho-\hat{\rho})\hat{t}^{\star},\epsilon
t^{\star},\epsilon\hat{t}^{\star}).\end{split}$ (53)

Also, by the same calculations, the angle at the wall may be approximated by:

$\begin{split}\theta^{w}(z^{\theta,\rho,\epsilon})&=\theta+\omega_{2}(I_{tan}(E)-\rho)\cdot(\frac{1}{2}T_{1}^{\rho}-t^{\star})+\mathcal{O}_{C^{r}}(\epsilon,\epsilon
t^{\star})\\\ &=\theta+\frac{1}{2}\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}(E)-\frac{1}{2}\rho\cdot\tau(I_{tan}(E))-\omega_{2}(I_{tan}(E))\cdot
t^{\star}\\\ &\qquad\qquad+\mathcal{O}_{C^{r}}(\epsilon,\rho^{2},\epsilon
t^{\star},\rho t^{\star})\end{split}$ (54)

Integrating backwards from $\bar{\theta}(z^{\theta,\rho,\epsilon})$ we obtain,
by the same arguments, and then by using (53):

$\begin{split}\theta^{w}(z^{\hat{\theta},\hat{\rho},\epsilon})&=\bar{\theta}(z^{\theta,\rho,\epsilon})-\omega_{2}(I_{tan}(E)-\hat{\rho})\cdot(\frac{1}{2}T_{1}^{\hat{\rho}}-\hat{t}^{\star})+\mathcal{O}_{C^{r}}(\epsilon,\epsilon\hat{t}^{\star})\\\
&=\bar{\theta}(z^{\theta,\rho,\epsilon})-\frac{1}{2}\omega_{2}(I_{tan}(E))\cdot
T_{1}^{tan}(E)+\frac{1}{2}\hat{\rho}\cdot\tau(I_{tan}(E))+\omega_{2}(I_{tan}(E))\cdot\hat{t}^{\star}\\\
&\qquad\qquad+\mathcal{O}_{C^{r}}(\epsilon,\hat{\rho}^{2},\epsilon\hat{t}^{\star},\hat{\rho}\hat{t}^{\star})\\\
&=\theta^{w}(z^{\theta,\rho,\epsilon})+\mathcal{O}_{C^{r-1}}(\epsilon,\rho^{2},\rho-\hat{\rho},(\rho-\hat{\rho})\hat{t}^{\star},\epsilon
t^{\star},\epsilon\hat{t}^{\star},\hat{\rho}\hat{t}^{\star})\end{split}$ (55)

So we indeed established the required forms. ∎

Next, we find the leading order behavior of $(t^{\star}+\hat{t}^{\star})$ and
establish that all the error terms in Eqs. (47) and (53) depend smoothly on
$\sqrt{\rho^{w}}$, the square root of the action-distance from tangency at the
wall.

######  Lemma 4.3.

For sufficiently small $\epsilon$, for any $z^{w}\in\Sigma^{w}_{E}$, with
$S_{2}(q_{2}^{w},p^{w}_{2})=(\theta^{w},I^{w}=I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})-\rho^{w})$
and small $\rho^{w}>0$, there exist unique flight times, $t^{\star}$ and
$\hat{t}^{\star}$ such that $z_{min}:=\Phi^{\epsilon}_{t^{\star}}z^{w}$ and
$\hat{z}_{min}:=\Phi^{\epsilon}_{-\hat{t}^{\star}}\mathcal{R}_{1}z^{w}$ are in
$\Sigma^{<}_{E}$. The flight times are smooth functions of
$\sqrt{\rho^{w}},\theta^{w}$,$E$ and $\epsilon$ and are of the form:

$t^{\star},\hat{t}^{\star}=\frac{\sqrt{2\rho^{w}\omega_{2}(I_{tan})}}{-V^{\prime}_{1}(q_{1}^{w})}(1+O_{C^{r-1}}(\epsilon,\sqrt{\rho^{w}})),$
(56)

namely, $\hat{t}^{\star}=t^{\star}(1+O_{C^{r-1}}(\epsilon,\sqrt{\rho^{w}}))$
(smoothness here is w.r.t. $z^{w}$).

###### Proof.

Since $z^{w}=(q_{1}^{w},p^{w}_{1},q^{w}_{2},p^{w}_{2})\in\Sigma^{w}_{E}$, the
iso-energy momentum, $p^{w}_{1}=-p^{\epsilon}_{1}(\rho^{w},\theta^{w})$, is
given by (the definition of $\rho^{w}$ implies that
$p^{\epsilon}_{1}(0,\theta^{w})=0$):

$\begin{split}p^{\epsilon}_{1}(\rho^{w},\theta^{w})&=\sqrt{2(E-V_{1}(q_{1}^{w})-\frac{(p^{w}_{2})^{2}}{2}-V_{2}(q^{w}_{2})-\epsilon
V_{c}(q_{1}^{w},q_{2}^{w}(\rho^{w},\theta^{w})))}\\\
&=\sqrt{2(E-V_{1}(q_{1}^{w})-H_{2}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})-\rho^{w})-\epsilon
V_{c}(q_{1}^{w},q^{w}_{2}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})-\rho^{w},\theta^{w}))}\\\
&=\sqrt{\rho^{w}}\sqrt{2(H_{2}^{\prime}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w}))+\epsilon\frac{\partial
V_{c}(q_{1}^{w},q^{w}_{2}(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w}),\theta^{w})))}{\partial
I})+O_{C^{r-1}}(\rho^{w})}\\\
&=\sqrt{\rho^{w}}\sqrt{2\omega_{2}(I_{tan}(E))+O_{C^{r-1}}(\rho^{w},\epsilon)}\\\
&\end{split}$

We see that $p^{\epsilon}_{1}(\rho^{w},\theta^{w},E)$ is, to leading order,
independent of $\theta^{w}$, it is $C^{r-1}$ smooth in $\theta^{w}$ and
$\frac{p^{\epsilon}_{1}(\rho^{w},\theta^{w},E)}{\sqrt{\rho^{w}}}$ is $C^{r-1}$
smooth in $\rho^{w},\theta^{w}$, namely,
$p^{\epsilon}_{1}(\rho^{w},\theta^{w})=\mathcal{O}_{C^{r-1}}(\sqrt{\rho^{w}},\epsilon)$.

As $\dot{p}_{1}=-V^{\prime}_{1}(q_{1}^{w})+O_{C^{r-1}}(\epsilon,z-z^{w})>0$
and $p_{1}$ is monotone increasing near $z^{w}$, for sufficiently small
$-p_{1}^{w}$ the first crossing time of $\Sigma_{E}^{<}$ (the section
$p_{1}=0,\dot{p}_{1}>0,q_{1}<q_{1}^{w}$), $t^{\star}$, is uniquely defined and
depends smoothly on i.c., namely on $z^{w}$. Since
$-p_{1}^{w}=\int_{0}^{t^{\star}}(-V^{\prime}_{1}(q_{1}^{w})+O_{C^{r-1}}(\epsilon,\Phi^{\epsilon}_{t}z^{w}-z^{w}))dt=-t^{\star}V^{\prime}_{1}(q_{1}^{w})(1+O_{C^{r}}(\epsilon,t^{\star}))$,
we conclude that

$t^{\star}=\frac{p^{w}_{1}}{V^{\prime}_{1}(q_{1}^{w})}(1+O_{C^{r}}(\epsilon,p_{1}^{w}))=\frac{\sqrt{2\rho^{w}\omega_{2}(I_{tan})}}{-V^{\prime}_{1}(q_{1}^{w})}(1+\mathcal{O}_{C^{r-1}}(\sqrt{\rho^{w}},\epsilon))$
(57)

Repeating the same computation for negative time for the initial condition
$\mathcal{R}_{1}z^{w}$ provides exactly the same results for
$\hat{t}^{\star}$, so, by Eq. (57),
$\hat{t}^{\star}=t^{\star}(1+\mathcal{O}_{C^{r-1}}(\sqrt{\rho^{w}},\epsilon))$.
∎

By uniqueness of solutions,
$t^{\star}(z^{w})=t^{\star}(z^{\theta,\rho,\epsilon})$ and
$\hat{t}^{\star}(z^{w})=\hat{t}^{\star}(z^{\theta,\rho,\epsilon})$. Hence, to
complete the proof we need to show that to leading order we can replace in
formula (56) $\rho^{w}$ by $\rho$ and that the error terms are small in
$\sqrt{\rho}$. To relate the smoothness w.r.t. $z^{w}$ to smoothness w.r.t. to
$\theta,\rho,\epsilon$ we establish:

######  Lemma 4.4.

For sufficiently small $\epsilon>0$, there exist $C^{r-2}$ functions
$G_{\star},G_{\star\star}$ such that
$\sqrt{\rho^{w}}(\sqrt{\rho},\theta;E,\epsilon)=\sqrt{\rho}(1+\epsilon
G_{\star}(\sqrt{\rho},\theta,E,\epsilon))$ and
$\hat{\rho}-\rho=\epsilon\rho^{w}G_{\star\star}(\sqrt{\rho^{w}},\theta,E,\epsilon)=O_{C^{r-2}}(\epsilon(\sqrt{\rho})^{2})$.

###### Proof.

Recall that
$z^{w}=\Phi^{\epsilon}_{t^{-,\epsilon}(z)}z^{\theta,\rho,\epsilon}=\mathcal{R}_{1}\Phi^{\epsilon}_{T_{1}^{\hat{\theta},\hat{\rho},\epsilon}-t^{+,\epsilon}(z)}z^{\hat{\theta},\hat{\rho},\epsilon}$
where $\Pi
z^{w}=(\theta^{w},I^{w}=I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})-\rho^{w})$.
Let $\Pi
z_{tan}^{w}=(I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w}),\theta^{w})$ denote
the wall point on the tangency curve with the same angle, $\theta^{w}$ as of
$z^{w}$. Since the tangency curve at the wall depends smoothly on
$\theta^{w}$, as is the flight time between $\Sigma_{E}$ and
$\Sigma_{E}^{>}\cup\Sigma_{E}^{<}$, the singularity-line backward angle is a
$C^{r}$ function of $\theta^{w}$ :
$\theta_{tan}(\theta^{w},\epsilon)=\theta(\Phi^{\epsilon}_{-t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}z_{tan}^{w})$.
By definition

$I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})=I_{tan}^{\epsilon}(\theta_{tan})+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt,\quad\theta^{w}=\theta(\Phi^{\epsilon}_{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}z^{\theta_{tan},0,\epsilon}).$

Since $z^{w}=\Phi^{\epsilon}_{t^{-,\epsilon}(z)}z^{\theta,\rho,\epsilon}$ we
also obtain:

${I^{w}}=I_{tan}^{\epsilon}(\theta)-\rho+\epsilon\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt$
(58)

so, as $\rho^{w}=I_{tan,\Sigma^{*}_{E}}^{\epsilon}(\theta^{w})-I^{w}$, we get:

$\begin{split}\rho^{w}&=\rho+I_{tan}^{\epsilon}(\theta_{tan})-I_{tan}^{\epsilon}(\theta)\\\
&\qquad+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt-\epsilon\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt,\end{split}$
(59)

Recall that
$z_{sm}^{\theta,\rho,\epsilon}(t)=z_{sm}^{\theta,0,\epsilon}(t)+\mathcal{O}_{C^{r}}(\rho)=z_{sm}^{\theta_{tan},0,\epsilon}(t)+\mathcal{O}_{C^{r}}(\rho,\theta-\theta_{tan})$,
and that its travel time to the transverse section $\Sigma_{E}^{<}$,
$t^{h,\epsilon}(z^{\theta,\rho,\epsilon})$, also depends smoothly on
$\theta,\rho$. Hence,

$\begin{split}{\rho^{w}-\rho}&=I_{tan}^{\epsilon}(\theta_{tan})-I_{tan}^{\epsilon}(\theta)\\\
&+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt-\epsilon\int_{0}^{t^{h,\epsilon}(z^{\theta,0,\epsilon})}\\{I,V_{c}\\}\mid_{z_{sm}^{\theta,0,\epsilon}(t)}dt+O_{C^{r}}(\epsilon\rho)\\\
&\qquad\qquad+\epsilon\int_{0}^{t^{\star}(\epsilon,\sqrt{\rho^{w}},\theta^{w})}\\{I,V_{c}\\}\mid_{\Phi^{\epsilon}_{t}(z^{w})}dt\\\
&=\epsilon\int_{0}^{t^{\star}(\epsilon,\sqrt{\rho^{w}},\theta^{w})}\\{I,V_{c}\\}\mid_{\Phi^{\epsilon}_{t}(z^{w})}dt+\epsilon(\theta-\theta_{tan}(\theta^{w}))\tilde{g}(\theta,\theta^{w},\epsilon)+O_{C^{r}}(\epsilon\rho)\\\
&=\epsilon
t^{\star}g_{1}(t^{\star},z^{w})+\epsilon(\theta-\theta_{tan}(\theta^{w}))g_{2}(\theta,\theta^{w},\epsilon)+O_{C^{r}}(\epsilon\rho)\end{split}$
(60)

where $\tilde{g},g_{1},g_{2}$ are $C^{r-1}$ functions of their arguments.
Notice that:

$\begin{split}\theta_{tan}(\theta^{w},\epsilon)&=\theta^{w}-\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}[\omega_{2}(I(\cdot))+\epsilon\\{\theta,V_{c}\\}]\mid_{z_{sm}^{\epsilon,0,\theta_{tan}}(t)}dt)\\\
&=\theta^{w}-\frac{1}{2}\omega_{2}(I_{tan}(E))T_{1}^{tan}(E)+O_{C^{r}}(\epsilon),\end{split}$
(61)

Hence, $\theta_{tan}^{\prime}(\theta^{w},\epsilon)=1+O_{C^{r}}(\epsilon)$,
and,

$\begin{split}\theta({\rho^{w}},\theta^{w},\epsilon)-\theta_{tan}&(\theta^{w},\epsilon)=\theta^{w}-\int_{0}^{t^{-,\epsilon}(z^{\theta,\rho,\epsilon})}[\omega_{2}(I(\cdot))+\epsilon\\{\theta,V_{c}\\}]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&\qquad\qquad-(\theta^{w}-\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}[]\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt)\\\
&=\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}[]\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt-\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{\star}}[]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&=\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}[]\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt-\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})}[]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t)}dt\\\
&\qquad\qquad+\int_{0}^{t^{\star}}[]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{\star}+t)}dt\\\
&=\int_{0}^{t^{h,0}(z^{\theta_{tan},0,0})}\omega_{2}(I(\cdot))\mid_{z_{sm}^{0,0,\theta_{tan}}(t)}dt-\int_{0}^{t^{h,0}(z^{\theta,\rho,0})}\omega_{2}(I(\cdot))\mid_{z_{sm}^{\theta,\rho,0}(t)}dt+O_{C^{r}}(\epsilon)\\\
&\qquad\qquad+\int_{0}^{t^{\star}}[]\mid_{z_{sm}^{\theta,\rho,\epsilon}(t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{\star}+t)}dt\\\
&=-\frac{1}{2}\rho\cdot\tau(I_{tan}(E))+\int_{0}^{t^{\star}}\omega_{2}(I(\cdot))\mid_{z_{sm}^{\theta,\rho,\epsilon}(t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{\star}+t)}dt+\mathcal{O}_{C^{r}}(\epsilon,\rho^{2},\epsilon
t^{\star})\\\
&=-\frac{1}{2}\rho\cdot\tau(I_{tan}(E))+\omega_{2}(I_{tan}(E))\cdot
t^{\star}+\mathcal{O}_{C^{r}}(\epsilon,\epsilon t^{\star},\rho
t^{\star},\rho^{2}).\end{split}$ (62)

so (60) and (62) become:

$\begin{split}{\rho}&+O_{C^{r}}(\epsilon\rho)=\rho^{w}-\epsilon
t^{\star}g_{1}(t^{\star},z^{w})-\epsilon(\theta-\theta_{tan}(\theta^{w}))g_{2}(\theta,\theta^{w},\epsilon)\\\
\theta&=\theta_{tan}(\theta^{w})+\omega_{2}(I_{tan}(E))\cdot
t^{\star}-\frac{1}{2}\rho\cdot\tau(I_{tan}(E))+\mathcal{O}_{C^{r}}(\epsilon,\epsilon
t^{\star},\rho t^{\star},\rho^{2}).\end{split}$ (63)

By proposition 1, the solution to this set of equations corresponds to the map
which maps uniquely an iso-energy impacting i.c. ($(\theta,\rho),\rho>0)$ to
an impacting phase-point on the wall ($(\rho^{w},\theta^{w}),\rho^{w}>0)$ and
a tangent i.c. ($(\theta,\rho)=(\theta_{tan}(\theta^{w}),0)$ to a phase point
on the wall which belongs to the tangency curve - the point
$(\rho^{w}=0,\theta^{w})$. At $\epsilon=0$, we have
${\rho}=\rho^{w},\theta=\theta_{tan}(\theta^{w})+\omega_{2}(I_{tan}(E))\cdot
t^{\star}-\frac{1}{2}\rho\cdot\tau(I_{tan}(E))+\mathcal{O}_{C^{r}}(\rho^{2})$.
By lemma 4.3,
$t^{\star}=\frac{\sqrt{2\rho^{w}\omega_{2}(I_{tan})}}{-V^{\prime}_{1}(q_{1}^{w})}(1+\mathcal{O}_{C^{r-1}}(\epsilon,\sqrt{\rho^{w}}))$,
so $\frac{\partial
t^{\star}}{\partial\theta^{w}}=\mathcal{O}_{C^{r-2}}(\epsilon\sqrt{\rho^{w}},\rho^{w})$
hence, for small $(\epsilon,\sqrt{\rho^{w}})$, by the Implicit function
Theorem for (63), viewed as a set of $C^{r-1}$ smooth implicit equations of
the form $F(\theta,\rho,\sqrt{\rho^{w}},\theta^{w};\epsilon)=0$, and noticing
that $\left|\frac{\partial
F}{\partial(\theta,\rho^{w})}\right|=\det\begin{pmatrix}1+O_{C^{r}}(\epsilon)&\mathcal{O}_{C^{r-1}}(\epsilon,\epsilon\sqrt{\rho^{w}})\\\
\frac{1}{2}\tau(I_{tan}(E))+\mathcal{O}_{C^{r-1}}(\sqrt{\rho^{w}},\rho)&-\theta_{tan}^{\prime}(\theta^{w},\epsilon)+\mathcal{O}_{C^{r-1}}(\epsilon\sqrt{\rho^{w}})\\\
\end{pmatrix}=-1+\mathcal{O}_{C^{r-1}}(\epsilon)$), there exist a smooth
$C^{r-1}$ function $\tilde{g}_{\star}$ such that

$\begin{split}{\rho}(\sqrt{\rho^{w}},\theta,E,\epsilon)&=\rho^{w}+\epsilon\sqrt{\rho^{w}}\tilde{g}_{\star}(\sqrt{\rho^{w}},\theta;E,\epsilon)\\\
&=\epsilon\sqrt{\rho^{w}}\tilde{g}_{\star}(0,\theta;E,\epsilon)+\rho^{w}(1+\epsilon\frac{\partial\tilde{g}_{\star}(0,\theta;E,\epsilon)}{\partial\sqrt{\rho^{w}}})+\mathcal{O}_{C^{r-3}}(\epsilon(\sqrt{\rho^{w}})^{3})\\\
\theta^{w}(\sqrt{\rho^{w}},\theta,E,\epsilon)&=\theta+\frac{1}{2}\omega_{2}(I_{tan}(E))T_{1}^{tan}(E)-\omega_{2}(I_{tan}(E))\cdot\frac{\sqrt{2\rho^{w}\omega_{2}(I_{tan})}}{-V^{\prime}_{1}(q_{1}^{w})}\\\
&\qquad+\mathcal{O}_{C^{r-1}}(\epsilon\sqrt{\rho^{w}},(\rho^{w})^{2})\end{split}$
(64)

and indeed $\rho$ and $\theta^{w}$ depend smoothly on $\sqrt{\rho^{w}}$ (but
not on $\rho^{w}$). Since the solution must map the upper half plane to the
upper half plane, and since for very small $\sqrt{\rho^{w}}$ the term
$\epsilon\sqrt{\rho^{w}}$ dominates
${\rho}(\sqrt{\rho^{w}},\theta,E,\epsilon)$, we conclude that for all $\theta$
and sufficiently small $\epsilon$,
$\tilde{g}_{\star}(0,\theta;E,\epsilon)\geqslant 0$. Thus, by area
preservation of the flow this term must vanish. Hence,
${\rho}(\sqrt{\rho^{w}},\theta,E,\epsilon)=\rho^{w}+\epsilon\rho^{w}g_{\star}(\sqrt{\rho^{w}},\theta;E,\epsilon)$
where $g_{\star}(\sqrt{\rho^{w}},\theta;E,\epsilon)$ is $C^{r-2}$, namely,
$\sqrt{\rho}(\sqrt{\rho^{w}},\theta;E,\epsilon)=\sqrt{\rho^{w}}(1+\frac{1}{2}\epsilon
g_{\star}(\sqrt{\rho^{w}},\theta;E,\epsilon)+\mathcal{O}_{C^{r-2}}(\epsilon^{2}))$
and $\frac{\partial(\sqrt{\rho})}{\partial\sqrt{\rho^{w}}}$ is close to one.
It follows that there exists an inverse function of the form:

$\sqrt{\rho^{w}}(\sqrt{\rho},\theta;E,\epsilon)=\sqrt{\rho}(1+\epsilon
G_{\star}(\sqrt{\rho},\theta,E,\epsilon)).$ (65)

Similarly, integrating $z^{\epsilon,\hat{\rho},\hat{\theta}}(t)$ up to
$\mathcal{R}_{1}z^{w}$ (recall that
$(I(\mathcal{R}_{1}z^{w}),\theta(\mathcal{R}_{1}z^{w}))=(I^{w},\theta^{w})$
since $\mathcal{R}_{1}$ denotes a reflection in the $(q_{1},p_{1})$ plain), we
obtain:

${I^{w}}=I_{tan}^{\epsilon}(\hat{\theta})-\hat{\rho}+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}_{\star}}\\{I,V_{c}\\}\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt$
(66)

so (60) becomes:

$\begin{split}\rho^{w}-\hat{\rho}&=I_{tan}^{\epsilon}(\theta_{tan})-I_{tan}^{\epsilon}(\hat{\theta})\\\
&+\epsilon\int_{0}^{t^{h,\epsilon}(z^{\epsilon,0,\theta_{tan}})}\\{I,V_{c}\\}\mid_{z_{sm}^{\epsilon,0,\theta_{tan}}(t)}dt-\epsilon\int_{0}^{t^{h,\epsilon}(z^{\epsilon,0,\hat{\theta}})}\\{I,V_{c}\\}\mid_{z_{sm}^{\epsilon,0,\hat{\theta}}(t)}dt+\mathcal{O}_{C^{r}}(\epsilon\hat{\rho})\\\
&\qquad\qquad-\epsilon\int_{0}^{\hat{t}^{\star}(\epsilon,\sqrt{\rho^{w}},\theta^{w})}\\{I,V_{c}\\}\mid_{\Phi^{\epsilon}_{-t}((\mathcal{R}_{1}z^{w})}dt\\\
&=-\epsilon\int_{0}^{\hat{t}^{\star}(\epsilon,\sqrt{\rho^{w}},\theta^{w})}\\{I,V_{c}\\}\mid_{\Phi^{\epsilon}_{-t}(\mathcal{R}_{1}z^{w})}dt+\epsilon(\hat{\theta}-\theta_{tan}(\theta^{w}))\hat{g}(\hat{\theta},\theta^{w},\epsilon)+\mathcal{O}_{C^{r}}(\epsilon\hat{\rho})\\\
&=\epsilon\hat{t}^{\star}\hat{g}_{1}(\hat{t}^{\star},z^{w})+\epsilon(\hat{\theta}-\theta_{tan}(\theta^{w}))\hat{g}_{2}(\hat{\theta},\theta^{w},\epsilon)+\mathcal{O}_{C^{r}}(\epsilon\hat{\rho})\end{split}$
(67)

and (62) becomes

$\begin{split}\hat{\theta}-\theta_{tan}&(\theta^{w},\epsilon)=\theta^{w}-\int_{0}^{t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}_{\star}}[\omega_{2}(I(\cdot))+\epsilon\\{\theta,V_{c}\\}]\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt\\\
&\qquad\qquad-(\theta^{w}-\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}[]\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt)\\\
&=\int_{0}^{t^{h,\epsilon}(z^{\theta_{tan},0,\epsilon})}[]\mid_{z_{sm}^{\theta_{tan},0,\epsilon}(t)}dt-\int_{0}^{t^{h,\epsilon}(z^{\hat{\theta},\hat{\rho},\epsilon})+\hat{t}^{\star}}[]\mid_{z_{sm}^{\hat{\theta},\hat{\rho},\epsilon}(t)}dt\\\
&\\\
&=\int_{0}^{t^{h,0}(z^{\theta_{tan},0,0})}\omega_{2}(I(\cdot))\mid_{z_{sm}^{\theta_{tan},0,0}(t)}dt-\int_{0}^{t^{h,0}(z^{\hat{\theta},\hat{\rho},0})}\omega_{2}(I(\cdot))\mid_{z_{sm}^{\hat{\theta},\hat{\rho},0}(t)}dt\\\
&\qquad\qquad-\int_{0}^{\hat{t}^{\star}}[]\mid_{\Phi^{\epsilon}_{-t}(\mathcal{R}_{1}z^{w})}dt+\mathcal{O}_{C^{r}}(\epsilon)\\\
&=-\frac{1}{2}\hat{\rho}\cdot\tau(I_{tan}(E))-\int_{0}^{\hat{t}^{\star}}\omega_{2}(I(\cdot))\mid_{\Phi^{\epsilon}_{-t}(\mathcal{R}_{1}z^{w})}dt+\mathcal{O}_{C^{r}}(\epsilon,\hat{\rho}^{2},\epsilon\hat{t}^{\star})\\\
&=-\frac{1}{2}\hat{\rho}\cdot\tau(I_{tan}(E))-\omega_{2}(I_{tan}(E))\cdot\hat{t}^{\star}+\mathcal{O}_{C^{r}}(\epsilon,\epsilon\hat{t}^{\star},\hat{\rho}\hat{t}^{\star},\hat{\rho}^{2}).\end{split}$
(68)

So, we obtain again implicit equations of the form (63) for
$(\hat{\rho},\hat{\theta},\sqrt{\rho^{w}},\theta^{w})$ and by the IFT (w.r.t
to $(\hat{\rho},\hat{\theta})$, we conclude that there exist unique smooth
solutions
$(\hat{\rho}(\sqrt{\rho^{w}},\theta^{w},\epsilon),\hat{\theta}(\sqrt{\rho^{w}},\theta^{w},\epsilon))$
and a function
$\tilde{\hat{g}}_{\star}(\sqrt{\rho^{w}},\theta^{w};E,\epsilon)$ such that
$\hat{\rho}(\sqrt{\rho^{w}},\theta^{w},E,\epsilon)=\rho^{w}+\epsilon\sqrt{\rho^{w}}\tilde{\hat{g}}_{\star}(\sqrt{\rho^{w}},\theta^{w};E,\epsilon)$.
Area preservation again implies that
$\tilde{\hat{g}}_{\star}(0,\theta^{w};E,\epsilon)$ must vanish for
sufficiently small $\epsilon$, so
$\hat{\rho}(\sqrt{\rho^{w}},\theta^{w},E,\epsilon)=\rho^{w}(1+\epsilon\hat{g}_{\star}(\sqrt{\rho^{w}},\theta^{w};E,\epsilon))$
for some $C^{r-2}$ function $\hat{g}_{\star}$. Hence, using the above
established solution of $(\theta,\rho^{w})$ as a function of
$(\sqrt{\rho^{w}},\theta)$, we conclude that
$\hat{\rho}-\rho=\epsilon\rho^{w}(\hat{g}_{\star}(\sqrt{\rho^{w}},\theta^{w}(\sqrt{\rho^{w}},\theta,E,\epsilon);E,\epsilon)-g_{\star}(\sqrt{\rho^{w}},\theta;E,\epsilon)):=\epsilon\rho^{w}G_{\star\star}(\sqrt{\rho^{w}},\theta,E,\epsilon)=O_{C^{r-2}}(\epsilon(\sqrt{\rho})^{2}).$
∎

Now we complete the proof of Proposition 4.1: By lemmas 4.3 and 4.4,
$t^{\star}+\hat{t}^{\star}=\frac{2\sqrt{2\rho^{w}\omega_{2}(I_{tan})}}{-V^{\prime}_{1}(q_{1}^{w})}(1+O_{C^{r-1}}(\epsilon,\sqrt{\rho^{w}}))=\frac{2\sqrt{2\rho\omega_{2}(I_{tan})}}{|V^{\prime}_{1}(q_{1}^{w})|}(1+O_{C^{r-2}}(\epsilon,\sqrt{\rho}))$.
Substituting this expression for $t^{\star}+\hat{t}^{\star}$ in (45) the
leading order behavior of (34b) is established. From the equations of motion
and lemma 4.4

$\begin{split}\theta-\hat{\theta}&=-\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})-t^{\star}}[\omega_{2}(I(\cdot))+\epsilon\\{\theta,V_{c}\\}]\mid_{\Phi^{\epsilon}_{-t}(z^{w})}dt+\int_{0}^{t^{h,\epsilon}(z^{\theta,\rho,\epsilon})+\hat{t}^{\star}}[]\mid_{\Phi^{\epsilon}_{-t}((\mathcal{R}_{1}z^{w})}dt\\\
&=\mathcal{O}_{C^{r}}(t^{\star},\hat{t}^{\star},|\Phi^{\epsilon}_{t^{\star}}z^{w}-\Phi^{\epsilon}_{-\hat{t}^{\star}}(\mathcal{R}_{1}z^{w})|)\\\
&=\mathcal{O}_{C^{r}}(t^{\star},\hat{t}^{\star},\sqrt{\rho^{w}})=\mathcal{O}_{C^{r-2}}(\sqrt{\rho}).\end{split}$
(69)

So, by lemma 4.4 the error terms in (45),
$\mathcal{O}_{C^{r}}(\rho\epsilon,\hat{\rho}\epsilon,\epsilon
t^{\star},\epsilon\hat{t}^{\star},(\theta-\hat{\theta})\epsilon)$ and
$\mathcal{O}_{C^{r}}(\epsilon,\rho^{2},\rho-\hat{\rho},(\rho-\hat{\rho})\hat{t}^{\star},\epsilon
t^{\star},\epsilon\hat{t}^{\star})$ all depend smoothly on $\sqrt{\rho}$ and
are of order $\mathcal{O}_{C^{r-2}}(\epsilon\sqrt{\rho})$ and
$\mathcal{O}_{C^{r-2}}(\epsilon,\rho^{2},\epsilon\sqrt{\rho})$ respectively,
so, by the definition of $\rho$, the error terms are as in (34b).

∎

###  4.2 Proof of Theorem 2

The final form of the local return map near the singularity line follows, by a
symplectic change of coordinates, from the form of
$\mathcal{F}_{\epsilon}^{loc}$ (Eq. (34)). Let
$S(\theta,K)=K\theta+\int^{\theta}I_{tan}^{\epsilon}(\theta^{\prime})d\theta^{\prime}$
denote a generating function of the second kind. Since
$I_{tan}^{\epsilon}(\theta)$ is a $C^{r}$ smooth function, $S(\theta,K)$ is
$C^{r+1}$. Then:

$\begin{split}I=\frac{\partial
S}{\partial\theta}=K+I_{tan}^{\epsilon}(\theta)\Rightarrow
K(\theta,I)&=I-I_{tan}^{\epsilon}(\theta)\\\ \phi(\theta,I)&=\frac{\partial
S}{\partial K}=\theta\end{split}$ (70)

i.e. this is a $C^{r}$ smooth symplectic transformation for which in the new
coordinates $(\phi,K)$, $K=0$ represents tangency in the perturbed system. The
image of $(\theta,I)$ under the local return map is
$(\bar{\theta},\bar{I})=\mathcal{F}^{loc}_{\epsilon}(\theta,I)$ and hence
$(\bar{\phi},\bar{K})=(\bar{\theta},\bar{I}-I_{tan}^{\epsilon}(\bar{\theta}))$,
so, by Equation (34), using (10) and the definition of $\Omega$ we find:

$K\geq
0:\begin{cases}\bar{K}=K-I_{tan}^{\epsilon}(\bar{\phi})+\bar{I}_{tan}^{\epsilon}(\bar{\phi})+\mathcal{O}_{C^{r-1}}(\epsilon
K)\\\
\bar{\phi}=\phi+\Omega+G_{s}(K)+\mathcal{O}_{C^{r-1}}(\epsilon,K^{2})\end{cases}$
(71a)
$K<0:\begin{cases}\bar{K}=K-I_{tan}^{\epsilon}(\bar{\phi})+\bar{I}_{tan}^{\epsilon}(\bar{\phi})+\mathcal{O}_{C^{r-2}}(\epsilon\sqrt{-K})\\\
\begin{aligned}
\bar{\phi}=\phi+\Omega&+G_{s}(K)+\mathcal{O}_{C^{r-2}}(\epsilon,K^{2},\epsilon\sqrt{-K})\end{aligned}\end{cases}$
(71b)

As, $\epsilon
f(\bar{\phi})=\bar{I}^{\epsilon}_{tan}(\bar{\phi})-I^{\epsilon}_{tan}(\bar{\phi})$
and $G_{s}(K)$ is of order $K$ for positive $K$ and of order $\sqrt{-K}$ for
negative $K$, the form (9) is established. ∎

##  5 Stable dynamics near tangency

The local return map of section 4 is valid for $I-I_{tan}(E)=o(1)$. For
$I-I_{tan}(E)=\mathcal{O}(1)$, it is known that stable, near integrable
behavior arises [23], namely, away from the singularity line orbits are
bounded away from being tangent and remain, forever (for 2 d.o.f. systems), in
either the impacting regime or the non-impacting regime. The intermediate
boundary layer, at which the local return map (9) has stable, near integrable
behavior is identified next: we show that for sufficiently small $\gamma>0$
and bounded $k_{s}$, initial conditions with $K=k_{s}\epsilon^{\gamma}$ do not
cross the singularity line at $K=0$ and their dynamics are near integrable.
The bounds on the values of $\gamma$ differ between the two sides of the
tangency curve; For any $(\alpha>0,\nu>0,\sigma\in\\{+1,-1\\})$, set

$\gamma_{c}(\Omega,\alpha,\nu,\sigma)=\begin{cases}1&\Omega\text{ is
$(\alpha,\nu)$-Diophantine, }\sigma=\pm 1\\\ \frac{1}{2}&\Omega\text{ is near
resonant (not $(\alpha,\nu)$-Diophantine), }\sigma=+1\\\
\frac{2}{3}&\Omega\text{ is near resonant (not $(\alpha,\nu)$-Diophantine),
}\sigma=-1\end{cases}$ (72)

where we recall that $(\alpha,\nu)$-Diophantine means that for all integers
$(k_{1},k_{2})$ with $k_{2}\neq 0$,

$|\frac{\Omega}{2\pi}-\frac{k_{1}}{k_{2}}|\geqslant\frac{\alpha}{|k_{2}|^{1+\nu}((\frac{k_{1}}{k_{2}})^{2}+1)^{\nu/2}}.$
(73)

######  Theorem 5.1.

For any $\Omega,\alpha,\nu>0,k_{s}\neq
0,\gamma\in(0,\gamma_{c}(\Omega,\alpha,\nu,sign(k_{s}))$, there exists
$\epsilon_{c}=\epsilon_{c}(k_{s},\gamma,\Omega,\alpha,\nu)>0$ such that for
any $\epsilon\in(0,\epsilon_{c})$, the return map (9) has an invariant KAM
circle which is at a distance $k_{s}\epsilon^{\gamma}$ from the singularity
line at $K=0$.

###### Proof.

Let $\tilde{k}_{s}=2\tau k_{s},\tilde{\epsilon}=\epsilon$ for $k_{s}>0$ and
$\tilde{k}_{s}=k_{s},\tilde{\epsilon}=\epsilon^{1-\gamma/2}$ for $k_{s}<0$. We
show below that when the near integrable smooth twist map:

$\begin{cases}\bar{I}=I+\tilde{\epsilon}\tilde{f}(\bar{\phi})\\\
\bar{\phi}=\phi+\Omega+I\end{cases}$ (74)

with a $C^{r}$ smooth $2\pi$-periodic forcing function
$\tilde{f}(\bar{\phi})=\tau f(\bar{\phi})$ has KAM rotational invariant curves
that are at a distance $\tilde{k}_{s}\tilde{\epsilon}^{\gamma}$ from $I=0$ for
all $\gamma\in(0,\gamma_{c})$ and
$\tilde{\epsilon}<\tilde{\epsilon}_{c}(\tilde{k}_{s},\Omega,\alpha,\nu)$, then
the return map (9) has also invariant KAM curves that are at a distance
$K=k_{s}\epsilon^{\gamma}$ from the singularity line (meaning that both the
minimal and maximal distances of the curve from $I=0$ are bounded from above
and below by some constants time $k_{s}\epsilon^{\gamma}$).

By standard KAM theory, the near identity smooth twist map (74) has such
invariant curves with $\gamma\in(0,1)$ for Diophantine $\Omega$ and
$\gamma\in(0,\frac{1}{2})$ for non-Diophantine $\Omega$; More precisely, for a
Diophantine $\Omega$ and $\tilde{\epsilon}<\tilde{\epsilon}_{c}$, there exist
a positive measure set $\Lambda_{\epsilon}$ of $I_{\epsilon}$ values in
$[-1,1]$, so that the KAM curves
$(I_{\Omega+I_{\epsilon}}^{\epsilon}(\phi),\phi)$ are invariant curves of the
map with rotation numbers $\Omega+I_{\epsilon}$, and there exists
$M=M(\Omega,\tilde{\epsilon}_{c})$ such that
$|I_{\Omega+I_{\epsilon}}^{\epsilon}(\phi)-I_{\epsilon}|\leqslant\tilde{\epsilon}M$.
In particular, for all ${\gamma}\in(0,1)$ and $\tilde{k}_{s}>0$, the upper set
$[\tilde{k}_{s}\tilde{\epsilon}^{\gamma}+\tilde{\epsilon}M,2\tilde{k}_{s}\tilde{\epsilon}^{\gamma}+\tilde{\epsilon}M]\cap\Lambda_{\epsilon}$
and the lower set
$[-\tilde{k}_{s}\tilde{\epsilon}^{\gamma}-\tilde{\epsilon}M,-2\tilde{k}_{s}\tilde{\epsilon}^{\gamma}-\tilde{\epsilon}M]\cap\Lambda_{\epsilon}$
include many invariant curves, and the corresponding invariant curves remain
$\pm\tilde{k}_{s}\tilde{\epsilon}^{\gamma}$ away from $I=0$ and at a maximal
distance of $\pm 2\tilde{k}_{s}\tilde{\epsilon}^{\gamma}\pm\tilde{\epsilon}M$
from the line $I=0$. For a non-Diophantine $\Omega$, averaging of the twist
map (74) brings it to its resonant normal form, and the resulting averaged map
has rotational KAM tori surrounding a resonance band of oscillatory orbits of
width $C(\Omega)\sqrt{\tilde{\epsilon}}$ around the origin. Rotational tori
that are at a distance larger than $\pm\tilde{k}_{s}\tilde{\epsilon}^{\gamma}$
with some $\gamma<\frac{1}{2}$ from $I=0$ belong to the Diophantine set of
rotational invariant curves of the normal form, and thus, as above, there
exists $M_{R}=M_{R}(\Omega,\tilde{\epsilon}_{c},\tilde{\gamma})$ such that
$|I_{\Omega+I_{\epsilon}}^{\epsilon}(\phi)-I_{\epsilon}|\leqslant\tilde{\epsilon}M_{R}$
for all
$I_{\epsilon}\in[\pm\tilde{k}_{s}\tilde{\epsilon}^{\gamma}\pm\tilde{\epsilon}M_{R},\pm
2\tilde{k}_{s}\tilde{\epsilon}^{\gamma}\pm\tilde{\epsilon}M_{R}]\cap\Lambda_{\epsilon}$,
namely, these tori remain at a distance
$\tilde{k}_{s}\tilde{\epsilon}^{\gamma}$ from the origin.

Since the return map (9) is $C^{r-2}$ smooth for positive $K$ values, for
positive $k_{s}$ the above observations suffice to prove the results; Indeed,
recall that for $K>0$, the function $G_{s}(K)$ of Eq. (10) is $C^{r}$ smooth
and vanishes at the origin. Thus, rescaling by $\frac{1}{\tau}$ and shifting
$K$ by $O(\epsilon)$ terms brings the map of Eq. (9) to a form which is
$C^{r-2}$ -close to the truncated map (74). Since $r>6$, if the truncated map
has invariant curves so does this map, and, moreover, the invariant curves of
these maps remain $\mathcal{O}_{C^{r-2}}(\epsilon I,\epsilon^{2})$-close to
each other. In particular, for sufficiently small $\epsilon$, for any
$\gamma>0$, the shifted rescaled map (9) has invariant curves with $I$ values
that remain at a distance $2\tau k_{s}\epsilon^{\gamma}$ away from $I=0$,
thus, provided $\gamma<1$, for sufficiently small $\epsilon$, $K$ remains at a
distance of at least $k_{s}\epsilon^{\gamma}$ from the singularity line (so
indeed all iterations remain in the smooth regime).

For negative $k_{s}$, let
$K(k)=k_{s}\epsilon^{\gamma}(1+\frac{\epsilon^{\beta}}{\tau k_{s}}k)$ so
$k=\epsilon^{-(\gamma+\beta)}\tau(K-k_{s}\epsilon^{\gamma})$ with
$0<\gamma<1,\beta>0$. We first derive the map for $(k,\phi)$ and establish
that for $k_{s}<0$ the resulting truncated map is $C^{r-2}$ close to the full
map, so that if the truncated map has KAM tori so does the full map. We then
rescale the truncated map to obtain the required form (74) with
$\tilde{\epsilon}=\epsilon^{1-\gamma/2}$, and the existence of KAM curves of
(74) for small shifted $I$ values completes the proof. In the resonant case,
the requirement that the shifted $I$ value is below the $\Omega$ resonance
band leads to the additional restriction on $\gamma$.

Plugging $K(k)$ in Eq. (9) leads to:

$\bar{k}=k+\epsilon^{1-(\gamma+\beta)}\tau
f(\bar{\phi})+\epsilon^{-(\gamma+\beta)}\tau\breve{g}(\epsilon G_{s}(K(k)))$
(75)

where $\breve{g}$ denotes a $C^{r-2}$ function that vanishes at the origin. We
next show that the last term is $C^{r-2}$-small (as a function of $k$). Recall
that $G_{s}(K)$ (see Eq. (10)) has a square root singularity for negative $K$,
so $\epsilon^{-(\gamma+\beta)}\tau\breve{g}(\epsilon
G_{s}(K(k)))=O(\epsilon^{1-(\gamma+\beta)}\sqrt{-K(k)})=O(\epsilon^{1-\frac{1}{2}\gamma-\beta})$,
and is indeed smaller by a factor of $\epsilon^{\frac{1}{2}\gamma}$ from the
leading order term. Moreover, while $\frac{d^{j}}{dK^{j}}G_{s}(K)$ at $K=K(k)$
is large, of order $\epsilon^{\gamma(\frac{1}{2}-j)}$, since
$K^{\prime}(k)=\frac{\epsilon^{\gamma+\beta}}{\tau}$, its derivatives w.r.t.
$k$ is small, of order $\epsilon^{\frac{1}{2}\gamma+j\beta}$, and the $j$th
order derivatives of $\epsilon^{-(\gamma+\beta)}\tau\breve{g}(\epsilon
G_{s}(K(k)))$ is, for $1\leqslant j\leqslant r-2$, small, of order
$\epsilon^{j-\frac{1}{2}\gamma+(j-1)\beta}$. Namely, we established

$\bar{k}=k+\epsilon^{1-(\gamma+\beta)}\tau
f(\bar{\phi})+O_{C^{r-2}}(\epsilon^{1-\frac{1}{2}\gamma-\beta}).$ (76)

Next, to find the leading order behavior for $\bar{\phi}$, we expand Eq. (10)
for negative $k_{s}$ and $k=O(1)$:

$\begin{split}G_{s}(k_{s}\epsilon^{\gamma}(1+\frac{\epsilon^{\beta}}{\tau
k_{s}}k))&=-\epsilon^{\gamma/2}\frac{\sqrt{|k_{s}|}(2\omega_{2}(I_{tan}(E)))^{3/2}}{|V_{1}^{\prime}(q_{1}^{w})|}\sqrt{1+\frac{\epsilon^{\beta}}{\tau
k_{s}}k}(1+O_{C^{r-2}}(K))\\\ &+\tau
k_{s}\epsilon^{\gamma}(1+\frac{\epsilon^{\beta}}{\tau
k_{s}}k)+O_{C^{r-2}}(K^{2})\\\
&=-\epsilon^{\gamma/2}\frac{\sqrt{|k_{s}|}(2\omega_{2}(I_{tan}(E)))^{3/2}}{|V_{1}^{\prime}(q_{1}^{w})|}\cdot(1+\frac{1}{2}\frac{\epsilon^{\beta}}{\tau
k_{s}}k+O_{C^{r-2}}(\epsilon^{2\beta},\epsilon^{\gamma},k\epsilon^{\gamma+\beta}))\\\
&+\tau k_{s}\epsilon^{\gamma}(1+\frac{\epsilon^{\beta}}{\tau
k_{s}}k)+O_{C^{r-2}}(\epsilon^{2\gamma},k\epsilon^{2\gamma+\beta})\\\
&=-\epsilon^{\gamma/2}\frac{\sqrt{|k_{s}|}(2\omega_{2}(I_{tan}(E)))^{3/2}}{|V_{1}^{\prime}(q_{1}^{w})|}+\tau
k_{s}\epsilon^{\gamma}+O_{C^{r-2}}(\epsilon^{3\gamma/2})\\\
&+k(\epsilon^{\gamma/2+\beta}\frac{\sqrt{|k_{s}|}(2\omega_{2}(I_{tan}(E)))^{3/2}}{-2\tau
k_{s}|V_{1}^{\prime}(q_{1}^{w})|}+\epsilon^{\gamma+\beta}+O_{C^{r-2}}(\epsilon^{3\gamma/2+\beta,2\gamma+\beta}))\end{split}$
(77)

Hence, we get

$\bar{\phi}=\phi+\hat{\Omega}_{\epsilon}+k(A\epsilon^{\gamma/2+\beta}+\epsilon^{(\gamma+\beta)}+O_{C^{r-2}}(\epsilon^{3\gamma/2}))+O_{C^{r-2}}(\epsilon^{3\gamma/2})$
(78)

where
$\Delta\Omega_{\epsilon}:=\hat{\Omega}_{\epsilon}-\Omega=-\epsilon^{\gamma/2}\frac{\sqrt{|k_{s}|}(2\omega_{2}(I_{tan}(E)))^{3/2}}{|V_{1}^{\prime}(q_{1}^{w})|}+O(\epsilon^{\gamma})$.
Summarizing, the map near $k_{s}\epsilon^{\gamma}$ is of the form:

$\begin{cases}\bar{k}=k+\epsilon^{1-\gamma-\beta}\tau
f(\bar{\phi})+O_{C^{r-2}}(\epsilon^{1-\frac{1}{2}\gamma-\beta})\\\
\bar{\phi}=\phi+\hat{\Omega}_{\epsilon}+kA\epsilon^{\gamma/2+\beta}+O_{C^{r-2}}(\epsilon^{\frac{3}{2}\gamma},k\epsilon^{\gamma+\beta})\end{cases}$
(79)

where
$A=\frac{(2\omega_{2}(I_{tan}(E)))^{3/2}}{2\tau\sqrt{|k_{s}|}|V_{1}^{\prime}(q_{1}^{w})|}$.
Hence, if the truncated map (the above map without the $O_{C^{r-2}}$ terms)
has KAM tori, then the map (79) has KAM curves which are
$O_{C^{r-2}}(\epsilon^{\min(1-\frac{1}{2}\gamma-\beta,\frac{3}{2}\gamma,\gamma+\beta)})$
close to them. Rescaling the truncated map by setting
$J=kA\epsilon^{\gamma/2+\beta}$ and shifting $J$ to
$I=J+\Delta\Omega_{\epsilon}$ we obtain the map (74) with
$\tilde{\epsilon}=\epsilon^{1-\frac{\gamma}{2}}$ (namely
$\epsilon=\tilde{\epsilon}^{\frac{2}{2-\gamma}}$, and
$\Delta\Omega_{\epsilon}=O(\epsilon^{\frac{\gamma}{2}})=O(\tilde{\epsilon}^{\frac{\gamma}{2-\gamma}})$).

For the Diophantine $\Omega$ case, for sufficiently small $\tilde{\epsilon}$,
(74) has a positive measure set of rotational KAM curves. In particular,
taking $\gamma\in(\frac{1}{2},1)$, we obtain
$\Delta\Omega_{\epsilon}<C\sqrt{\tilde{\epsilon}}$, so for sufficiently small
$\tilde{\epsilon}$ there is a positive measure set of rotational KAM tori near
$I_{0}=\Delta\Omega_{\epsilon}$, and the corresponding shifted $J$ values
satisfy
$|J_{KAM}(\phi,J_{0})-J_{0}|<C_{1}\tilde{\epsilon}=C_{1}\epsilon^{1-\gamma/2}$.
Taking $J_{0}$ values of order
$\epsilon^{\gamma/2+\beta}=\tilde{\epsilon}^{\frac{\gamma+2\beta}{2-\gamma}}\ll|\Delta\Omega_{\epsilon}$|,
the corresponding $k$ values satisfy
$|k^{trun}_{KAM}(\phi,k_{0})-k_{0}|<C_{2}\epsilon^{1-\gamma-\beta}$ for the
truncated map. The KAM curves of (79), $k_{KAM}(\phi,k_{0})$, are
$O_{C^{r-2}}(\epsilon^{\min(1-\frac{1}{2}\gamma-\beta,\frac{3}{2}\gamma,\gamma+\beta)})$
-close to the invariant curves of the truncated map (even though they may be
supported on a different set of $k_{0}$ values), so
$|\epsilon^{\beta}k_{KAM}(\phi,k_{0})-\epsilon^{\beta}k_{0}|<C_{2}\epsilon^{\min(1-\gamma,\frac{3}{2}\gamma+\beta,\gamma+2\beta)}\ll
1$ (recall that $\gamma<1,\beta>0$), hence, there is a positive measure set of
order one $k_{0}$ values such that
$K_{KAM}(\phi,k_{0})=k_{s}\epsilon^{\gamma}(1+\epsilon^{\beta}k_{KAM}(\phi,k_{0}))$
are invariant curves which are strictly below the singularity line:

$|K_{KAM}(\phi,k_{0};k_{s},\gamma,\beta,\Omega_{D})-k_{s}\epsilon^{\gamma}|<k_{s}\epsilon^{\gamma+\beta}(k_{0}+\epsilon^{\min(1-\frac{1}{2}\gamma-\beta,\frac{3}{2}\gamma,\gamma+\beta)}).$
(80)

We conclude that these curves correspond to rotational KAM curves of (9) that
bound the tangency band from below at a distance $k_{s}\epsilon^{\gamma}$ from
the singularity line for $\gamma$ arbitrary close to the critical exponent
$\gamma_{c}(\Omega_{Diophantine},\alpha,\nu,-1)=1$, which also implies that
other rotational invariant curves with smaller $\gamma$ values remain below
that tangency band.

For the resonant $\Omega$ case, for
$\gamma<\frac{2}{3}=\gamma_{c}(\Omega_{Resonance},\alpha,\nu,-1)$, the shift
in the action,
$|\Delta\Omega_{\epsilon}|=O(\tilde{\epsilon}^{\frac{\gamma}{2-\gamma}})>C\sqrt{\tilde{\epsilon}}$,
and thus, for sufficiently small $\tilde{\epsilon}$ there is again a positive
measure set of rotational KAM tori near $I_{0}=\Delta\Omega_{\epsilon}$.
Following exactly the same steps as in the Diophantine case with negative
$k_{s}$, we conclude that the map (9) has invariant curves of the form (80)
for all $\gamma<\frac{2}{3}$. ∎

Theorem 5.1 implies that the tangency band boundaries,
$I^{\epsilon}_{\pm}(\theta)=K^{\epsilon}_{\pm}(\theta)+I_{tan}^{\epsilon}(\theta)$
(see Def. 2.5) are at most a distance $k_{s}\epsilon^{\gamma}$ from
$I_{tan}^{\epsilon}(\theta)$ which is order $\epsilon$ close to $I_{tan}$,
thus implies Theorem 3. From the form of the map (9) it is clear that for
$K=o(\epsilon)$ small or even $K=0$, the next iteration in the return map is
of $\mathcal{O}(\epsilon)$, and hence a lower bound on the width of the
tangency band is $\mathcal{O}(\epsilon)$.

##  6 The truncated tangency map

To demonstrate the stability results of Theorem 5.1 and to support claims 2.6
-2.7 we derive first some basic properties of the truncated tangency map,
$F_{TTM}$ of Eq. (12). This map corresponds to taking the leading order terms
of $G_{s}(K)$ in the map (9). Notice that in (9) $\alpha>0$ and, by rescaling
$K$, we can take $\tau\in\\{\pm 1\\}$; rescaling $K$ by $|\tau(I_{tan}(E))|$
shows that the parameters $(\alpha,\epsilon,\tau)$ in (12) are related to
those of (9) by :
$\alpha\leftarrow\frac{(2\omega_{2}(I_{tan}(E)))^{3/2}}{\sqrt{|\tau(I_{tan}(E))|}|V_{1}^{\prime}(q_{1}^{w})|}>0$,
$\epsilon\leftarrow\epsilon|\tau(I_{tan}(E))|,\tau\leftarrow\frac{\tau(I_{tan}(E))}{|\tau(I_{tan}(E))|}\
\in\\{\pm 1\\}$ and $K\leftarrow|\tau(I_{tan}(E))|K$ (recall that we study
here the behavior near a regular torus which is tangent to the wall, so
$V_{1}^{\prime}(q_{1}^{w}),\omega_{2}(I_{tan}(E)),\tau(I_{tan}(E))\neq 0$).
The symmetry property of Lemma 3.8 shows that
$f(\bar{\phi})=\bar{I}_{tan}^{\epsilon}(\bar{\phi})-I_{tan}^{\epsilon}(\bar{\phi})$
is an odd function of $\bar{\phi}$. We first establish a few properties of
$F_{TTM}$ and then, taking the first non-trivial Fourier mode of $f$, we set
$f({\phi})=\sin\phi$ and study numerically the resulting map, establishing
claims 2.6 -2.8 for this specific realization.

###  6.1 Twist and fixed points of the truncated tangency map

The twist of the unperturbed truncated tangency map (Eq. (12) at $\epsilon=0$)
is piecewise smooth:

$\frac{\partial\bar{\phi}}{\partial K}=\begin{cases}\tau&K>0\\\
\tau+\frac{\alpha}{2\sqrt{-K}}&K<0\end{cases}$ (81)

Hence, for $\tau>0$ the map $F_{TTM}$ is a piecewise smooth twist map: it has
a strictly positive twist for all values of $K$. For $\tau<0$, the twist
changes sign and a unique non-twist circle is created at $K<0$ (on the
impacting side):

$\tau+\frac{\alpha}{2\sqrt{-K_{NT}}}=0\Rightarrow
K_{NT}=-\frac{\alpha^{2}}{4\tau^{2}}.$ (82)

For $K$ values which are bounded away from $0,K_{NT}$ the twist is bounded
away from $0$. For both positive and negative shear, the twist diverges when
the singularity line is approached from below, at $K\rightarrow 0^{-}$. The
stability results of section 5 utilize this twist property for circles which
are bounded away from the singularity set and twist-less circle.

The limits of small and large $\alpha$ are interesting and deserve a separate
study - recall that
$\alpha=\frac{(2\omega_{2}(I_{tan}(E)))^{3/2}}{\sqrt{|\tau(I_{tan}(E))|}|V_{1}^{\prime}(q_{1}^{w})|}$,
so large $\alpha$ appears when
$\sqrt{|\tau(I_{tan}(E))|}|V_{1}^{\prime}(q_{1}^{w})|$ is small - either when
the wall is approaching an elliptic fixed point and then $\alpha$ is large for
all energies, or, when the energy is such that the tangent torus is close to a
twist-less torus of the smooth system.

The fixed points of the map, as listed in Appendix B and summarized in the
bifurcation diagram in the $(\Omega,K)$ plane presented in Figure 4, appear,
for each zero $\phi^{*}$ of $f(\phi)$, as a countable set of continuous
branches that emanate from $K=0,\Omega=2\pi j,\>\ j\in\mathbb{Z}$. For
positive shear these branches are monotone deceasing in $\Omega$. They
correspond to unstable fixed points when $f^{\prime}(\phi^{*})>0$. When
$f^{\prime}(\phi^{*})<0$ they correspond to centers, with the exception of
order $\epsilon$ interval of negative $K$ values at which the fixed point is a
saddle with negative multipliers. For negative shear, each branch includes a
saddle node bifurcation at $K=K_{NT}$, so there is a range of $\Omega$ values
at which each branch contributes three fixed points, two of which with nearby
$K$ values.

Figure 4:  Bifurcation diagrams for the map (12). First column
$f^{\prime}(\phi^{*})>0$, second column $f^{\prime}(\phi^{*})<0$. First two
rows $\alpha=1,$ last row $\alpha=2\sqrt{8\pi}$. First row $\tau=1$, second
and third rows $\tau=-1$. In all figures $\epsilon=0.3$. Notice that the
branches of centers at negative $K^{*}$ values undergo period-doubling
bifurcation at small $\Omega-2\pi j$ values (second column).

When the fixed points spacing in $K$ becomes small, we expect chaotization of
the map by three main mechanisms: overlap of resonance regions around stable
fixed points, heteroclinic tangles of nearby saddles and sequences of period
doubling bifurcations. Such proximity occurs in two situations: first, for
$\tau=-1$, at $\Omega=2\pi j+\delta$, the positive and negative branches of
the fixed points are $\delta$ close (a singular saddle-center bifurcation, the
positive branch is at $K=\delta$ whereas the negative branch is close to
$-\frac{\delta^{2}}{\alpha^{2}}$), so near the bifurcation point complex
dynamics is always observed. Second, in the large $\alpha$ limit the negative
branches are close to each other and to the singularity line, and undergo, as
$\epsilon$ is increased, sequences of period doubling bifurcations. For
$\alpha\gg\sqrt{8\pi}\approx 5.01$ and small
$l\ll\left\lfloor\frac{\alpha^{2}}{8\pi}\right\rfloor$, we observe that the
spacing between the first $l$ branches is of order $\frac{1}{\alpha^{2}}$.
Overlap between $\sqrt{\epsilon}$ resonances around stable points is thus
expected when $\epsilon>\frac{C}{\alpha^{4}}$. Moreover, these branches become
unstable when $\epsilon>C\frac{2\pi l+\Omega\text{ mod }2\pi}{\alpha^{2}}$,
for some constant $C$, so for such values of $\epsilon$ sequences of period
doubling bifurcations occur close to the singularity line. These observations
may explain the fast mixing which is observed at high $\alpha$ values (see
Figures .

###  6.2  The dynamics in the tangency band

Figures 5 demonstrates the main claim of Theorem 5.1: for small $\epsilon$,
KAM curves with strictly positive $K$ values and KAM curves with strictly
negative $K$ values are observed for both Diophantine and resonant rotation
numbers $\Omega$. The tangency band, $\mathcal{B}_{\epsilon}$ is bounded in
between these curves. On the other hand, as is demonstrated in Figures 6-15,
which show zoom-ins and longer trajectory segments in the tangency band,
trajectories that pass close the singularity line do not lie on rotational
invariant curves.

Tangency band width (Claim 2.6): Figure 6 shows the dependence of the maximal
upper and lower widths of the tangency band on $\epsilon$. We iterate 100
tangent initial conditions for $10^{5}$ iterates, plotting, for each
$\epsilon$, $\max(K):=\max_{\phi_{0},j}{K_{j}}\leqslant\
W^{+},\min(K)=\min_{{\phi_{0},j}}{K_{j}}\geqslant-W^{-}$ where
$j=1,...,10^{5}$, $K_{0}=0$ and $\phi_{0}\in[-\pi,\pi]$ is evenly spaced (in
the next subsections we study the dynamics and the width of individual
trajectories in the band and provide a tighter numerical bound on the band
widths). Figure 6a,b summarises the findings: for regular $\Omega$, $W^{\pm}$
are linear in $\epsilon$ and are essentially symmetric. Figure 6c shows that
for $\Omega=2\pi$ the $\sqrt{\epsilon}$ scaling of $W^{+}$ and the
$\epsilon^{2/3}$ of $W^{-}$ suggested from Theorem 5.1 are realized. On the
other hand, Figure 6a,b demonstrate that the resonant $\Omega$ values which
are not strongly resonant produce $W^{\pm}$ which appear to be linear in
$\epsilon$, with widths close to those of the nearby irrational $\Omega$.
The above scalings are valid for sufficiently small $\epsilon$: as $\epsilon$
is increased, the tangency band overlaps with other resonances, reminiscent of