Differentiability of the Largest Lyapunov Exponent for Non-Planar Open Billiards

Abstract

This paper investigates the behaviour of open billiard systems in high-dimensional spaces. Specifically, we estimate the largest Lyapunov exponent, which quantifies the rate of divergence between nearby trajectories in a dynamical system. This exponent is shown to be continuous and differentiable with respect to a small perturbation parameter. A theoretical analysis forms the basis of the investigation. Our findings contribute to the field of dynamical systems theory and have significant implications for the stability of open billiard systems, which are used to model physical phenomena. The results provide a deeper comprehension of the behaviour of open billiard systems in high-dimensional spaces and emphasise the importance of taking small perturbations into consideration when analysing these systems.

1. Introduction

Billiards represent dynamic systems where a particle travels at a constant speed and collides with the boundary of the billiard’s domain following the principles of geometrical optics, which state that “the angle of incidence equals the angle of reflection”. Open billiards are a specialised category of billiard systems that occur within unbounded regions. In these scenarios, the domain encompasses the area outside a finite number of strictly convex compact obstacles. These obstacles satisfy the no-eclipse condition (H) established by Ikawa [1]. This condition ensures that the convex hull formed by any pair of obstacles does not intersect with any other obstacle, effectively preventing the existence of a straight line that passes through more than two obstacles. This condition preserves the smoothness and predictability of the open billiard system by ensuring that trajectories within the system avoid tangent points and singularities. This simplifies the analysis and understanding of the system’s behaviour, allowing for a clearer study of particle motion within the open billiard system. One way to quantify the behaviour of this system is through the use of Lyapunov exponents. Lyapunov exponents measure the rate at which neighbouring trajectories in a dynamical system diverge or converge, revealing the growth or decay of small perturbations. In the case of open billiards, the hyperbolic non-wandering set of the billiard map indicates the presence of positive and negative Lyapunov exponents. Numerous studies have investigated Lyapunov exponents for billiards, (see [2,3,4,5,6,7]). In this paper, we continue our investigation of the regularity properties of Lyapunov exponents from [8] for open billiards in Euclidean spaces. In [8], we studied the case of open billiards in the plane; here we deal with the higher-dimensional case.

Our primary findings are as follows:

The largest Lyapunov exponent for an open billiard in ${\mathbb{R}}^n$ is

$${\lambda }_1=\underset{m\to \infty }{lim}\frac{1}{m}\sum _{i=1}^{m}log\left(1+d_i\left(0\right){\ell }_i\left(0\right)\right),$$

where $d_i$ is the distance between two reflection points along trajectories, and ${\ell }_i$ is related to the curvature of the unstable manifold, as defined in (2).

To comprehend the following theorems, an understanding of non-planar billiard deformations and their concepts is required. A small deformation parameter $\alpha \in [0,b],b\in \mathbb{R}$ characterises billiard deformations, which include position, rotation, and obstacle reshaping. The initial boundary parameterisation and deformation parameter $\alpha$ are used to parameterise the obstacle borders. This was introduced in [9]. Billiard deformations and notations in the subsequent theorems are explained in Section 4. This study assumes that billiard deformation is a differentiable function for both parameters, providing the foundation for the theorems.

Theorem 1. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{4,1}$ billiard deformation in ${\mathbb{R}}^n,n\ge 3$. Let ${\lambda }_1\left(\alpha \right)$ be the largest Lyapunov exponent for $K\left(\alpha \right)$. Then, the largest Lyapunov exponent is continuous as a function of α.

Theorem 2. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{5,2}$ billiard deformation in ${\mathbb{R}}^n,n\ge 3$. Let ${\lambda }_1\left(\alpha \right)$ be the largest Lyapunov exponent for $K\left(\alpha \right)$. Then ${\lambda }_1\left(\alpha \right)$ is ${\mathcal{C}}^1$ with respect to α.

The proofs of these theorems are in Section 7 and Section 8, respectively.

The demonstration of the continuity and differentiability of the largest Lyapunov exponent for open billiards is highly significant. Open billiards represent a chaotic physical system (see, e.g., [1,10,11]), which appears naturally in scattering theory and tomography. Unlike many systems, which often exhibit complex and unpredictable behaviour in response to perturbations, open billiards show a more regular response, offering insights into the predictability of chaotic systems’ dynamics. This understanding is critical to improving our understanding of complex systems and their applications.

2. Preliminaries

2.1. Open Billiards

Let $K_1,K_2,\dots ,K_{z_0}$ be strictly convex, disjoint, and compact subsets of ${\mathbb{R}}^n,n\ge 3$ with smooth boundaries $\partial K_i$, and satisfying condition (H) of Ikawa [1]: for any $i\ne j\ne k$ the convex hull of $K_i\cup K_k$ does not have any common points with $K_j$. Let $\Omega$ be the exterior of K (i.e., $\Omega =\overline{{\mathbb{R}}^n\setminus K}$), where $K={\bigcup }_i{K}_i$. Let

$$\mathcal{M}=\{x=(q,v)\in int(\Omega )\times {\mathbb{S}}^{n-1}or(q,v)\in \partial \Omega \times {\mathbb{S}}^{n-1}:⟨{\nu }_{\partial K}\left(q\right),v⟩\ge 0\},$$

where $n_{\partial K}\left(q\right)$ is the outwards unit normal vector to $\partial K$ at q. For $t\in \mathbb{R}$ and $x\in \mathcal{M}$, the billiard flow ${\Phi }_t$ is defined as ${\Phi }_t\left(x\right)=(q_t,v_t)$, where $q_t$ and $v_t$ represent the position and velocity of the x at time t. Let $\Lambda$ represent the set of all points of $\mathcal{M}$ that have bounded trajectories. Let

$$M=\{x=(q,v)\in (\partial K\times {\mathbb{S}}^{n-1})\cap \mathcal{M}:⟨\nu \left(q\right),v⟩\ge 0\}.$$

Let $t_j\in \mathbb{R}$, where $j\in \mathbb{Z}$, represent the time of j-th reflection. The billiard map $B:M\to M$ is defined as $B(q_0,v_0)=(q_1,v_1)$ where $q_1=q_0+t_1{v}_0\in \partial K_j$ and $v_1=v_0-2⟨v_0,\nu \left(q_1\right)⟩\nu \left(q_1\right)$. Clearly, B is a smooth diffeomorphism on M. Define the canonical projection map $\pi :M\to \partial K$ by $\pi (q,v)=q$. Let $M_0\subset M$ be the non-wandering set of the billiard map B, that is, $M_0=\{x\in M: |t_j\left(x\right)| <\infty ,forallj\in \mathbb{Z}\}$. It is clear that $M_0$ is invariant under B. See [3,4,12,13,14] for general information about billiard dynamical systems.

2.2. Symbolic Coding for Open Billiards

Each particular $x\in M_0$ can be coded by a bi-infinite sequence

$$\xi \left(x\right)=(\dots ,{\xi }_{-1},{\xi }_0,{\xi }_1,\dots )\in {\{1,2,\dots ,z_0\}}^{\mathbb{Z}},$$

in which ${\xi }_i\ne {\xi }_{i+1}$, for all $i\in \mathbb{Z}$, and ${\xi }_j$ indicates the obstacle $K_{{\xi }_j}$ such that $\pi B^j\left(x\right)\in \partial K_{{\xi }_j}$. For example, if there are three obstacles $K_1,K_2,K_3$ and $K_4$ as above and a particular x repeatedly hits $K_2,K_1,K_4,K_3,K_2,K_1,K_4,K_3$, then the bi-infinite sequence is $(\dots ,2,1,4,3,2,1,4,3,\dots )$. Let $\Sigma$ be the symbol space, which is defined as:

$$\Sigma =\{\xi =(\dots ,{\xi }_{-1},{\xi }_0,{\xi }_1,\dots )\in {\{1,2,\dots ,z_0\}}^{\mathbb{Z}}:{\xi }_i\ne {\xi }_{i+1},\forall i\in \mathbb{Z}\}.$$

Define the representation map $R:M_0\to \Sigma$ by $R\left(x\right)=\xi \left(x\right)$. Let $\sigma :\Sigma \to \Sigma$ be the two-sided subshift map defined by $\sigma \left(\xi \right)={\xi }^{\prime }$ where ${\xi }_i^{\prime }={\xi }_{i+1}$.

It is known that the representation map $R:M_0\to \Sigma$ is a homeomorphism (see, e.g., [12]). See [1,10,11,12,15], for topics related to symbolic dynamics for open billiards.

2.3. Lyapunov Exponents

For the open billiard $B:M_0⟶M_0$ in ${\mathbb{R}}^n$ we will use the coding $R:M_0⟶\Sigma$ from Section 2.2, which conjugates B with the shift map $\sigma :\Sigma ⟶\Sigma$, to define Lyapunov exponents. It follows from the symbolic coding that there are ergodic $\sigma$-invariant measures $\mu$ on $\Sigma$. Let $\mu$ be an ergodic $\sigma$-invariant probability measure on $\Sigma$. The following is a consequence of Oseledets Multiplicative Ergodic Theorem (see, e.g., [16,17]):

Theorem 3 (A Consequence of Oseledets Multiplicative Ergodic Theorem).

There exist real numbers ${\lambda }_1>{\lambda }_2>\dots >{\lambda }_k>0>-{\lambda }_k>\dots >-{\lambda }_1$ and vector subspaces $E_j^u\left(x\right)$ and $E_j^s\left(x\right)$ of $T_x\left(M\right)$, $1\le j\le k\le n-1$, $x\in M_0$, depending measurably on $R\left(x\right)\in \Sigma$ such that:

1. 

$E^u\left(x\right)=E_1^u\left(x\right)\oplus \dots \oplus E_k^u\left(x\right)$ and $E^s\left(x\right)=E_1^s\left(x\right)\oplus \dots \oplus E_k^s\left(x\right)$ for almost all $x\in M$;

2. 

$D_{x}B\left(E_i^u\left(x\right)\right)=E_i^u\left(B\left(x\right)\right)$ and $D_{x}B\left(E_i^s\left(x\right)\right)=E_i^s\left(B\left(x\right)\right)$ for all $x\in M_0$ and all $i=1,\dots ,k$, and

3. 

For almost all $x\in M_0$ there exists

$$\underset{m\to \infty }{lim}\frac{1}{m}log\parallel D_x{B}^m\left(w\right)\parallel ={\lambda }_j$$

whenever $w\in E_j^u\left(x\right)\oplus \dots \oplus E_k^u\left(x\right)$ ($j\le k$), however, $w\notin E_{j+1}^u\left(x\right)\oplus \dots \oplus E_k^u\left(x\right)$.

And

$$\underset{m\to \infty }{lim}\frac{1}{m}log\parallel D_x{B}^m\left(w\right)\parallel =-{\lambda }_j$$

whenever $w\in E_j^s\left(x\right)\oplus \dots \oplus E_k^s\left(x\right)$ ($j\le k$), however, $w\notin E_{j+1}^s\left(x\right)\oplus \dots \oplus E_k^s\left(x\right)$.

Here, “for almost all x” means “for almost all x” with respect to $\mu$. The numbers ${\lambda }_1,\dots ,{\lambda }_k$ are called Lyapunov exponents, while the invariant subspaces $E_j\left(x\right)$ are called Oseledets subspaces.

2.4. Propagation of Unstable Manifolds for Open Billiards

This part explains the relationship between unstable manifolds for the billiard ball map and the billiard flow in ${\mathbb{R}}^n$. Recall that $M_0$ represents the non-wandering set of the billiard ball map, while $\Lambda$ represents the non-wandering set of the billiard flow. For the billiard map, the unstable manifolds of size $ϵ$ are

$$W_{ϵ}^u\left(x\right)=\{y\in M:d(B^{-n}\left(x\right),B^{-n}\left(y\right))\le ϵ\mathrm{for}\mathrm{all}n\in \mathbb{N},d(B^{-n}\left(x\right),B^{-n}\left(y\right)){\to }_{n\to \infty }0\}.$$

Similarly, for $x=(q,v)\in \Lambda$ the unstable manifolds ${\tilde{W}}_{ϵ}^u\left(x\right)$ for the billiard ball flow are

$${\tilde{W}}_{ϵ}^u\left(x\right)=\{y\in \mathcal{M}:d({\Phi }_t\left(x\right),{\Phi }_t\left(y\right))\le ϵ\mathrm{for}\mathrm{all}t\le 0,d({\Phi }_t\left(x\right),{\Phi }_t\left(y\right)){\to }_{t\to -\infty }0\}.$$

It is known that the unstable manifolds for the billiard ball map and the billiard flow naturally are related. This correspondence can be described as follows geometrically. Given a point $x_0=(q_0,v_0)$ in the non-wandering set $M_0$, and a small number $0<t^{\left(1\right)}<t_1$, let $y_0=(q_0+t^{\left(1\right)}{v}_0,v_0)$. Then, there exists a one-to-one correspondence map ${\tilde{\phi }}_1$ between the unstable manifold $W^u\left(x_0\right)$ and the unstable manifold ${\tilde{W}}^u\left(y_0\right)$. In addition, it follows from Sinai [13,14], that the unstable manifold ${\tilde{W}}^u\left(y_0\right)$ has the form

$${\tilde{W}}^u\left(y_0\right)=\{(p,n_{Y_0}\left(p\right)):p\in Y_0\},$$

where $Y_0$ is a smooth $(n-1)$-dimensional hypersurface in ${\mathbb{R}}^n$ containing the point $y_0$ and is strictly convex with respect to the unit normal field $n_{Y_1}\left(p\right)$.

Likewise, for all $j=1,2,\dots ,m$, there exists one-to-one correspondence ${\tilde{\phi }}_j$ between the unstable manifolds $W^u\left(x_j\right)$ and ${\tilde{W}}^u\left(y_j\right)$, where $B^j\left(x_0\right)=x_j=(q_j,v_j)$ and $y_j=(q_j+t^{\left(j\right)}{v}_j,v_j)$, for a small positive $t_{j-1}<t^{\left(j\right)}<t_j-t_{j-1}$. Moreover, ${\tilde{W}}^u\left(y_j\right)$ takes the form $\widetilde{Y_j}=\{(p_j,n_{Y_j}\left(p_j\right));j\in Y_j\}$, where $Y_j$ is also a smooth $(n-1)$-dimensional hypersurface in ${\mathbb{R}}^n$ containing the point $y_j$ and is strictly convex with respect to the unit normal field $n_{Y_j}\left(p_j\right)$.

The following are the commutative diagrams involving the unstable manifolds for open billiard maps and for the billiard flows, as shown above:

$$\begin{array}{ccccccccc}{W}^u\left(x_0\right)& \stackrel{B\left(x_0\right)}{⟶}& W^u\left(x_1\right)& \stackrel{B^2\left(x_0\right)}{⟶}& W^u\left(x_2\right)& \dots \dots & W^u\left(x_{m-1}\right)& \stackrel{B^m\left(x_0\right)}{⟶}& W^u\left(x_m\right)\\ \downarrow {\tilde{\phi }}_0& & \downarrow {\tilde{\phi }}_1& & \downarrow {\tilde{\phi }}_2& & \downarrow {\tilde{\phi }}_{m-1}& & \downarrow {\tilde{\phi }}_m\\ {\tilde{W}}^u\left(y_0\right)& \stackrel{{\Phi }_{{\tau }_1}\left(y_0\right)}{⟶}& {\tilde{W}}^u\left(y_1\right)& \stackrel{{\Phi }_{{\tau }_2}\left(y_0\right)}{⟶}& {\tilde{W}}^u\left(y_2\right)& \dots \dots & {\tilde{W}}^u\left(y_{m-1}\right)& \stackrel{{\Phi }_{{\tau }_m}\left(y_0\right)}{⟶}& {\tilde{W}}^u\left(y_m\right)\end{array}$$

where ${\tau }_j=t_j+t^{(j+1)}$. Similarly, the following are the commutative diagrams involving the corresponding tangent spaces of unstable manifolds under the derivative of the billiard ball maps and the derivative of the billiard flows:

$$\begin{array}{ccccccccc}{E}^u\left(x_0\right)& \stackrel{DB}{⟶}& E^u\left(x_1\right)& \stackrel{D{B}^2}{⟶}& E^u\left(x_2\right)& \dots \dots & E^u\left(x_{m-1}\right)& \stackrel{D{B}^m}{⟶}& E^u\left(x_m\right)\\ \downarrow D{\tilde{\phi }}_0& & \downarrow D{\tilde{\phi }}_1& & \downarrow D{\tilde{\phi }}_2& & \downarrow D{\tilde{\phi }}_{m-1}& & \downarrow D{\tilde{\phi }}_m\\ {\tilde{E}}^u\left(y_0\right)& \stackrel{D{\Phi }_{{\tau }_1}}{⟶}& {\tilde{E}}^u\left(y_1\right)& \stackrel{D{\Phi }_{{\tau }_2}}{⟶}& {\tilde{E}}^u\left(y_2\right)& \dots \dots & {\tilde{E}}^u\left(y_{m-1}\right)& \stackrel{D{\Phi }_{{\tau }_m}}{⟶}& {\tilde{E}}^u\left(y_m\right)\end{array}$$

where $E^u\left(x_j\right)=T_{x_j}\left(W^u\left(x_j\right)\right)$ and ${\tilde{E}}^u\left(y_j\right)=T_{y_j}\left({\tilde{W}}^u\left(y_j\right)\right)$, and ${\tilde{E}}^u\left(y_j\right)\perp v_j$. Since the derivatives $D{\tilde{\phi }}_{j-1}$ and $D{\tilde{\phi }}_j$, for all j, are uniformly bounded [13], then there exist global constants $C>c>0$ such that

$$c\parallel D{\Phi }_{{\tau }_m}\parallel \le \parallel D{B}^m\parallel \le C\parallel D{\Phi }_{{\tau }_m}\parallel .$$

(1)

This will be used to calculate the largest Lyapunov exponent ${\lambda }_1$ for the open billiard map in Section 3.

Based on the prior discussion, we now apply the concept of unstable manifold propagation to write the main theorem, which involves the propagation of an appropriate convex curve on a convex hypersurface. This theorem was proved in reference [18] and is utilised to calculate the largest Lyapunov exponent, ${\lambda }_1$.

Let $x=(q,v)\in M_0$ and let $W_{ϵ}^u\left(x_0\right)$ be the local unstable manifold for $x_0$ for sufficiently small $ϵ>0$. Then, $W_{ϵ}^u\left(x_0\right)=\{(x,n_X\left(x\right)):x\in X\}$, where X is a convex curve on a smooth hypersurface $\tilde{X}$ containing q such that $\tilde{X}$ is strictly convex with respect to the unit normal field $n_{\tilde{X}}$. This follows from (cf. [13,14]), see also [18]. Let ${\mathcal{K}}_x:T_q\tilde{X}\to T_q\tilde{X}$ be the curvature operator (second fundamental form). Since $\tilde{X}$ is strictly convex, then the curvature ${\mathcal{K}}_x$ is positive definite with respect to the unit normal field $n_{\tilde{X}}$.

Let X be parameterised by $q\left(s\right),s\in [0,a]$, such that $q\left(0\right)=q_0+r{v}_0$ for a small $r>0$, and by the unit normal field $n_X\left(q\left(s\right)\right)$. Let $q_j\left(s\right),j=1,2,\dots ,m$ be the j-th reflection points of the forward billiard trajectory $\gamma \left(s\right)$ generated by $x\left(s\right)=(q\left(s\right),n_X\left(q\left(s\right)\right)$. We assume that $a>0$ is sufficiently small so that the j-th reflection points $q_j\left(s\right)$ belong to the same boundary component $\partial K_{{\xi }_j}$ for every $s\in [0,a]$. Let $0<t_1\left(x\left(s\right)\right)<\dots <t_m\left(x\left(s\right)\right)$ be the times of the reflections of the ray $\gamma \left(s\right)$ at $\partial K$. Let ${\kappa }_j\left(s\right)$ be the curvature of $\partial K$ at $q_j\left(s\right)$ and ${\varphi }_j\left(s\right)$ be the collision angle between the unit normal ${\nu }_j\left(s\right)$ to $\partial K$ and the reflection ray of $\gamma \left(s\right)$ at $q_j\left(s\right)$. Also, let $d_j\left(s\right)$ be the distance between two reflection points, i.e., $d_j\left(s\right)=\parallel q_{j+1}\left(s\right)-q_j\left(s\right)\parallel$, $j=0,1,\dots ,m$.

Given a large $m\ge 1$, let $t_m\left(x\left(s\right)\right)<t<t_{m+1}\left(x\left(s\right)\right)$. Set $\widehat{X}=\{(q\left(s\right),n_X\left(s\right)):s\in [0,a]\}$, and ${\Phi }_t\left(\widehat{X}\right)={\widehat{X}}_t$. Let $\pi \left({\Phi }_t\left(x\left(s\right)\right)\right)=p\left(s\right)$. Then $p\left(s\right),s\in [0,a]$, is a parameterisation of the ${\mathcal{C}}^3$ curve $X_t=\pi \left({\Phi }_t\left(\widehat{X}\right)\right)$.

Let $k_j\left(s\right)$ be the normal curvature of $X_{t_j\left(s\right)}={lim}_{t\searrow t_j\left(s\right)}{X}_t$ at $q_j\left(s\right)$ in the direction ${\widehat{w}}_j\left(s\right)$$(\parallel {\widehat{w}}_j\left(s\right)\parallel =1)$ of ${lim}_{t\searrow t_j\left(s\right)}(d/d{s}^{\prime })w_t{\left(s^{\prime }\right)}_{|s^{\prime }=s}$ where ${\widehat{w}}_j\left(s\right)={\dot{q}}_t\left(s\right)/\parallel {\dot{q}}_t\left(s\right)\parallel$. For $j\ge 0$ let

$${\mathcal{K}}_j\left(s\right):T_{q_j\left(s\right)}\left(X_{t_j\left(s\right)}\right)\to T_{q_j\left(s\right)}\left(X_{t_j\left(s\right)}\right)$$

be the curvature operator of $X_{t_j\left(s\right)}$ at $q_j\left(s\right)$, and define $l_j\left(s\right)>0$ by

$${\left(1+d_j\left(s\right){\ell }_j\left(s\right)\right)}^2=1+2d_j\left(s\right)k_j\left(s\right)+{\left(d_j\left(s\right)\right)}^2{\parallel {\mathcal{K}}_j\left(s\right){\widehat{w}}_j\left(s\right)\parallel }^2.$$

(2)

Set

$${\delta }_j\left(s\right)=\frac{1}{1+d_j\left(s\right){\ell }_j\left(s\right)},1\le j\le m.$$

(3)

Theorem 4 ([18]). 

For all $s\in [0,a]$ we have

$$\parallel \dot{q}\left(s\right)\parallel =\parallel \dot{p}\left(s\right)\parallel {\delta }_1\left(s\right){\delta }_2\left(s\right)\dots {\delta }_m\left(s\right).$$

(4)

2.5. Curvature of Unstable Manifolds

Following [13,14,19] (see also [20]), here we express the curvature operator ${\mathcal{K}}_j$ of the convex front $X_j$ at $q_j\left(s\right)$ using certain related objects:

• ${\tilde{X}}_j^{-}$ represents the convex front passing $q_j\left(s\right)$ before collision, that is ${\tilde{X}}_j^{-}={lim}_{\tau \nearrow t_j}{\tilde{X}}_{\tau }$, where $t_{j-1}<\tau <t_j$.
• ${\tilde{X}}_j^{+}$ represents the convex front passing $q_j\left(s\right)$ after collision, that is ${\tilde{X}}_j^{+}={lim}_{\tilde{\tau }\searrow t_j}{\tilde{X}}_{\tilde{\tau }}$, where $t_j<\tilde{\tau }<t_{j+1}$. We write ${\tilde{X}}_j$ to indicate to ${\tilde{X}}_j^{+}$.
• ${\mathcal{J}}^{-}$ is the hyperplane of ${\tilde{X}}_j^{-}$ at $q_j\left(s\right)$, i.e., ${\mathcal{J}}^{-}=T_{q_j}\left({\tilde{X}}_j^{-}\right)$, which is perpendicular to $v_{j-1}$.
• ${\mathcal{J}}^{+}$ is the hyperplane of ${\tilde{X}}_j^{+}$ at $q_j\left(s\right)$, i.e., ${\mathcal{J}}^{+}=T_{q_j}\left({\tilde{X}}_j^{+}\right)$, which is perpendicular to $v_j$.
• $\mathcal{T}$ is the hyperplane of $\partial K$ at $q_j\left(s\right)$, i.e., $\mathcal{T}=T_{q_j}(\partial K)$, which is perpendicular to ${\nu }_j\left(q_j\left(s\right)\right)$.

Now, the curvature operator ${\mathcal{K}}_j:{\mathcal{J}}^{+}\to {\mathcal{J}}^{+}$ is given by

$${\mathcal{K}}_j=U_j^{-1}{\mathcal{K}}_j^{-}{U}_j+2\cos {\varphi }_j{V}_j^{\ast }{N}_j{V}_j,$$

(5)

where

• ${\mathcal{K}}_j^{-}$ is the curvature operator of ${\tilde{X}}_j^{-}$, which defined as ${\mathcal{K}}_j^{-}={\mathcal{K}}_{j-1}{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}$ or we can write ${\mathcal{K}}_j^{-}={({\mathcal{K}}_{j-1}^{-1}+d_{j-1}I)}^{-1}$.
• The unitary operator $U_j:{\mathcal{J}}_j^{+}\to {\mathcal{J}}_j^{-}$ is a projection parallel to ${\nu }_j\left(q_j\left(s\right)\right)$, defined as; for all $w_j\in {\mathcal{J}}_j^{+}$

  $$U_j{w}_j=w_j-\frac{⟨w_j,v_{j-1}⟩}{\cos {\varphi }_j}{\nu }_j.$$

  (6)
• $U_j^{-1}:{\mathcal{J}}_j^{-}\to {\mathcal{J}}_j^{+}$ is a projection parallel to ${\nu }_j\left(q_j\left(s\right)\right)$, defined as; for all $w_{j-1}\in {\mathcal{J}}_j^{-}$

  $$U_j^{-1}{w}_{j-1}=w_{j-1}-\frac{⟨w_{j-1},v_j⟩}{\cos {\varphi }_j}{\nu }_j.$$

  (7)
• $V_j:{\mathcal{J}}_j^{+}\to {\mathcal{T}}_j$ is a projection parallel to $v_j$, defined as; for all $w_j\in {\mathcal{J}}_j^{+}$

  $$V_j{w}_j=w_j-\frac{⟨w_j,{\nu }_j⟩}{\cos {\varphi }_j}{v}_j.$$

  (8)
• $V_j^{\ast }:{\mathcal{T}}_j\to {\mathcal{J}}_j^{+}$ is a projection parallel to ${\nu }_j\left(q\left(s\right)\right)$, defined as; for all ${\tilde{u}}_j\in {\mathcal{T}}_j$

  $$V_j^{\ast }{\tilde{u}}_j={\tilde{u}}_j-\frac{⟨{\tilde{u}}_j,v_j⟩}{\cos {\varphi }_j}{\nu }_j.$$

  (9)
• $N_j$ is the curvature operator s.f.f. of $\partial K$ at $q_j\left(s\right)$.

The operator $\parallel V^{\ast }NV\parallel$ is bounded by ${\kappa }_{min}\le \parallel V^{\ast }NV\parallel \le \frac{{\kappa }_{\max }}{{\cos }^2{\varphi }_j}$ where ${\kappa }_{min}$ and ${\kappa }_{\max }$ are the minimum and maximum eigenvalues of the normal curvature N and again ${\varphi }_j\left(s\right)$ is the collision angle at $q_j\left(s\right)$ and ${\varphi }_j\left(s\right)\in [0,\pi /2]$. Let ${\mu }_j\left(s\right)={\mu }_j\left(x\left(s\right)\right)$ and ${\eta }_j\left(s\right)={\eta }_j\left(x\left(s\right)\right)$ be the eigenvalues of the curvature operator ${\mathcal{K}}_j$. Then by using these and (5), we obtain

$${\mu }_{min}\le 2{\kappa }_{min}\le {\mu }_{j+1}\left(s\right)\le {\eta }_{j+1}\left(s\right)\le \frac{{\eta }_j\left(s\right)}{1+d_j\left(s\right){\eta }_j\left(s\right)}+2\frac{{\kappa }_{\max }}{\cos {\varphi }_j\left(s\right)}\le \frac{1}{d_{min}}+2\frac{{\kappa }_{\max }}{\cos {\varphi }_{\max }}\le {\eta }_{\max }.$$

(10)

3. Estimation of the Largest Lyapunov Exponent for Non-Planar Open Billiards

Here, we want to estimate the largest Lyapunov exponent ${\lambda }_1$ for non-planar open billiards. We use Oseledets multiplicative ergodic Theorems 3 and 4.

Assume that $\mu$ is an ergodic $\sigma$-invariant measure on $\Sigma$, and let $x=(q,v)\in M_0$ correspond to a typical point in $\Sigma$ with respect to $\mu$ via the representation map R. As in Theorem 3, there exists a subset $A_0$ of $\Sigma$ with $\mu \left(A_0\right)=1$ such that

$${\lambda }_1=\underset{m\to \infty }{lim}\frac{1}{m}log\parallel D_x{B}^m\left(w\right)\parallel ,$$

with $w\in E_1^u\left(x\right)\oplus E_2^u\left(x\right)\oplus \cdots \oplus E_k^u\left(x\right)\setminus E_2^u\left(x\right)\oplus E_3^u\left(x\right)\oplus \cdots \oplus E_k^u\left(x\right)$.

Let $w=\dot{q}\left(s\right)$ as in Section 4. And then by using (1), there exist some global constants $c_1>c_2>0$, independent of $x_0$, X, m, etc., such that

$$c_2\parallel \dot{p}\left(s\right)\parallel \le \parallel D_{x_0}{B}^m\left(w\right)\parallel \le c_1\parallel \dot{p}\left(s\right)\parallel$$

for all $s\in [0,a]$. So, by (4),

$$\frac{c_2}{{\delta }_1\left(0\right){\delta }_2\left(0\right)\dots {\delta }_m\left(0\right)}\le \parallel D_{x_0}{B}^m\left(w\right)\parallel \le \frac{c_1}{{\delta }_1\left(0\right){\delta }_2\left(0\right)\dots {\delta }_m\left(0\right)}$$

for all $s\in [0,a]$. Using this for $s=0$, taking logarithms and limits as $m\to \infty$, we obtain

$$-\underset{m\to \infty }{lim}\frac{1}{m}log\left({\delta }_1\left(0\right){\delta }_2\left(0\right)\dots {\delta }_m\left(0\right)\right)\le \underset{m\to \infty }{lim}\frac{1}{m}\parallel D_{x_0}{B}^m\left(w\right)\parallel \le -\underset{m\to \infty }{lim}\frac{1}{m}log\left({\delta }_1\left(0\right){\delta }_2\left(0\right)\dots {\delta }_m\left(0\right)\right).$$

Hence,

$$\begin{array}{cc}\hfill {\lambda }_1& =\underset{m\to \infty }{lim}-\frac{1}{m}\sum _{i=1}^{m}log{\delta }_i\left(0\right)=\underset{m\to \infty }{lim}\frac{1}{m}\sum _{i=1}^{m}log\left(1+d_i\left(0\right){\ell }_i\left(0\right)\right).\hfill \end{array}$$

(11)

We can use (2) and the (10) to estimate the upper and lower of the largest Lyapunov exponent ${\lambda }_1$ as follows:

$${\left(1+d_j\left(s\right){\ell }_j\left(s\right)\right)}^2\le 1+2d_j\left(s\right){\eta }_j\left(s\right)+{\left(d_j\left(s\right)\right)}^2{\left({\eta }_j\left(s\right)\right)}^2={\left(1+d_j\left(s\right){\eta }_j\left(s\right)\right)}^2.$$

(12)

This implies that

$$1+d_j\left(s\right){\ell }_j\left(s\right)\le 1+d_j\left(s\right){\eta }_j\left(s\right)\le 1+d_{\max }{\eta }_{\max }.$$

(13)

In the same way, we obtain

$$1+d_j\left(s\right){\ell }_j\left(s\right)\ge 1+d_j\left(s\right){\mu }_j\left(s\right)\ge 1+d_{min}{\mu }_{min}.$$

Therefore,

$$log(1+d_{min}{\mu }_{min})\le {\lambda }_1\le log(1+d_{\max }{\eta }_{\max }).$$

4. Billiard Deformations in ${\mathbb{R}}^{\mathit{n}}$

Let $\alpha \in I=[0,b]$, for some $b\in {\mathbb{R}}^{+}$, be a deformation parameter and let $\partial K_j\left(\alpha \right)$ be parameterised counterclockwise by ${\phi }_j(u_j^{\left(1\right)},u_j^{\left(2\right)},\dots ,u_j^{(n-1)},\alpha )$. Let $q_j={\phi }_j(u_j^{\left(1\right)},u_j^{\left(2\right)},\dots ,$$u_j^{(n-1)},\alpha )$ be a point that lies on $\partial K_j\left(\alpha \right)$. Denote the perimeter of $\partial K_i\left(\alpha \right)$ by $L_j\left(\alpha \right)$, and let $R_j=\{(u_j^{\left(1\right)},u_j^{\left(2\right)},\dots ,u_j^{(n-1)}:\alpha \in I,u_j^{\left(t\right)}\in [0,L_j\left(\alpha \right)]\}$.

Definition 1 ([9]). 

For any $\alpha \in I=[0,b]$, let $K\left(\alpha \right)$ be a subset of ${\mathbb{R}}^n,n\ge 3$. For integers $r\ge 2,r^{\prime }\ge 1$, we call $K\left(\alpha \right)$ a ${\mathcal{C}}^{r,r^{\prime }}$-billiard deformation if the following conditions hold for all $\alpha \in I$:

1. 

$K\left(\alpha \right)={\bigcup }_{i=1}^{z_0}{K}_i\left(\alpha \right)$ satisfies the no-eclipse condition $\left(\mathbf{H}\right)$.

2. 

Each $\partial K_i\left(\alpha \right)$ is a compact, strictly convex set with ${\mathcal{C}}^r$ boundary, and $K_i\left(\alpha \right)\bigcap K_j\left(\alpha \right)=\varphi$ for $i\ne j$.

3. 

For each $i=1,2,\dots ,m$ and all $p\in \partial K_i\left(\alpha \right)$, there is a rectangle $R_p\subset {\mathbb{R}}^{n-1}$ and a ${\mathcal{C}}^{r,r^{\prime }}$ function ${\phi }_j:R_p\to {\mathbb{R}}^n$, which is an orthonormal parametrisation of $\partial K_i\left(\alpha \right)$ at p.

4. 

For all integers $0\le l\le r,0\le l^{\prime }\le r^{\prime }$ (apart from $l=l^{\prime }=0$), there exist constants $C_{\phi }^{(l,l^{\prime })}$ depending only on the choice of the billiard deformation and the parameterisation ${\phi }_j$, such that for all integers $j=1,2,3,\dots ,z_0$,

$${\displaystyle \parallel }\frac{{\partial }^{l^{\prime }}}{\partial {\alpha }^{l^{\prime }}}{\nabla }_i^l{\phi }_j{\displaystyle \parallel }\le C_{\phi }^{(l,l^{\prime })}.$$

We consider the open billiard deformation map, denoted as $B_{\alpha }$, defined on the non-wandering set $M_{\alpha }$ for $K\left(\alpha \right)$. As in Section 2.2, we define ${\Sigma }_{\alpha }$ and $R_{\alpha }$ as the mapping from $M_{\alpha }$ to ${\Sigma }_{\alpha }$ such that $R_{\alpha }\left(x\left(\alpha \right)\right)=\xi \left(x\left(\alpha \right)\right)$. Using the parameterisation defined earlier, we can express the point corresponding to deformed billiard trajectories as $q_{{\xi }_j}\left(\alpha \right)={\phi }_{{\xi }_j}(u_j^{\left(1\right)}\left(\alpha \right),u_j^{\left(2\right)}\left(\alpha \right),\dots ,u_j^{(n-1)}\left(\alpha \right),\alpha )\in \partial K_{{\xi }_j}\left(\alpha \right)$, where $u_{{\xi }_j}^{\left(t\right)}\left(\alpha \right)\in [0,L_{{\xi }_j}\left(\alpha \right)]$. We write $q_j\left(\alpha \right)={\phi }_j(u_j^{\left(1\right)}\left(\alpha \right),u_j^{\left(2\right)}\left(\alpha \right),\dots ,u_j^{(n-1)}\left(\alpha \right),\alpha )$ for brevity.

It was shown in [9] that $u_j^{\left(t\right)}\left(\alpha \right)=u_{{\xi }_j}^{\left(t\right)}\left(\alpha \right)$, where $t=1,2,\dots ,n-1$, for a fixed $\xi \in {\Sigma }_{\alpha }$, is differentiable with respect to $\alpha$ and its higher derivative is bounded by a constant independent of $\alpha$ and j.

Theorem 5. 

[9] Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation with $r\ge 2,r^{\prime }\ge 1$. Then for all $t=1,2,\dots ,n-1$, $u_j^{\left(t\right)}\left(\alpha \right)$ is ${\mathcal{C}}^{min\{r-1,r^{\prime }\}}$ with respect to α, and there exist constants $C_u^{\left(s^{\prime }\right)}>0$ such that

$${\displaystyle \parallel }\frac{{\partial }^{s^{\prime }}{u}_j^{\left(t\right)}\left(\alpha \right)}{\partial {\alpha }^{s^{\prime }}}{\displaystyle \parallel }\le C_u^{\left(s^{\prime }\right)}.$$

5. Propagation of Unstable Manifolds for Non-Planar Billiard Deformations

Here, we want to re-describe the propagation of the unstable manifold mentioned in Section 4 for the billiard deformation $K\left(\alpha \right)$ with respect to the deformation parameter $\alpha \in I=[0,b]$. Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$, with $r\ge 4,r^{\prime }\ge 1$. Let $\tilde{X}$ be a smooth hypersurface passing through $x_0\left(\alpha \right)=(q_0\left(\alpha \right),v_0\left(\alpha \right))\in M_{\alpha }$ and let X be a ${\mathcal{C}}^3$ convex curve, with respect to the unit normal field ${\nu }_X$, on $\tilde{X}$. Let X be parameterised by $q(s,\alpha )$ for all $s\in [0,a]$ and set ${\widehat{X}}_{\alpha }=\{(q(s,\alpha ),{\nu }_X\left(q(s,\alpha )\right)),\forall s\in [0,a]\}$. For $j=1,2,\dots ,m$, let $q_j(s,\alpha )$ be the reflection points generated by $x(s,\alpha )=\left(q\right(s,\alpha ),\nu (q(s,\alpha )\left)\right)$, $d_j(s,\alpha )$ be the distance between $q_j(s,\alpha )$ and $q_{j+1}(s,\alpha )$, and ${\varphi }_j(s,\alpha )$ be the angle of reflection at $q_j(s,\alpha )$.

Let $t_{m-1}\left(x(s,\alpha )\right)<t<t_m\left(x(s,\alpha )\right)$ for some large $m\ge 2$. Let $\pi \left({\Phi }_t\left({\widehat{X}}_{\alpha }\right)\right)=X_{{\alpha }_t}.$ Then, $X_{{\alpha }_t}$ is a ${\mathcal{C}}^3$ curve on ${\tilde{X}}_t$ parameterised by $u_t(s,\alpha )$.

Let ${\mathcal{K}}_j(s,\alpha )$ be the curvature operator of $X_{\alpha t_j\left(s\right)}={lim}_{t\searrow t_j\left(x\left(s\right)\right)}{X}_{{\alpha }_t}$ at $q_j(s,\alpha )$ in the direction ${\widehat{w}}_j(s,\alpha )$ of ${lim}_{t\searrow t_j(\left(x\left(s\right)\right)}(d/d{s}^{\prime }){\widehat{w}}_t(s^{\prime },{\alpha }^{\prime }){)}_{\left|\right(s,\alpha )}$. Let ${\mathcal{K}}_0\left(s\right)$ be the curvature operator of X at $q(s,\alpha )$, which is independent of $\alpha$. And, for $j\ge 1$ let

$${\mathcal{K}}_j(s,\alpha ):T_{q_j(s,\alpha )}\left(X_{{\alpha }_{t_j\left(x\left(s\right)\right)}}\right)\to T_{q_j(s,\alpha )}\left(X_{{\alpha }_{t_j\left(x(s,\alpha )\right)}}\right)$$

be the curvature operator of $X_{t_j\left(x(s,\alpha )\right)}$ at $q_j(s,\alpha )$. Define ${\ell }_j(s,\alpha )$ by

$${\left(1+d_j(s,\alpha ){\ell }_j(s,\alpha )\right)}^2={\parallel w_j(s,\alpha )+d_j(s,\alpha ){\mathcal{K}}_j(s,\alpha )\parallel }^2,\mathrm{for}\mathrm{all}j=0,1,\dots ,m.$$

(14)

Then

$${\left(1+d_j(s,\alpha ){\ell }_j(s,\alpha )\right)}^2=1+2d_j(s,\alpha )k_j(s,\alpha )+{\left(d_j(s,\alpha )\right)}^2{\parallel {\mathcal{K}}_j\left(s\right){\widehat{w}}_j\left(s\right)\parallel }^2,$$

(15)

where $k_j(s,\alpha )$ is the normal curvature of $X_{{\alpha }_j}(s,\alpha )$ at $q_j(s,\alpha )$ in the direction ${\widehat{w}}_j(s,\alpha )$, which is given by $k_j(s,\alpha )=k_j\left({\widehat{w}}_j(s,\alpha )\right)=⟨{\mathcal{K}}_j(s,\alpha ){\widehat{w}}_j(s,\alpha ),{\widehat{w}}_j(s,\alpha )⟩$.

Now, we want to re-write the curvature operator in (5) with respect to the billiard deformation parameter $\alpha$. So, we have for $j=1,2,\dots ,m$

$${\mathcal{K}}_j(s,\alpha )=U_j^{-1}(s,\alpha ){({\mathcal{K}}_{j-1}^{-1}(s,\alpha )+d_{j-1}(s,\alpha )I)}^{-1}{U}_j(s,\alpha )+2\cos {\varphi }_j(s,\alpha )V_j^{\ast }(s,\alpha )N_j(s,\alpha )V_j(s,\alpha )$$

(16)

For brevity, we will write all previous characteristics as, e.g., $\mathcal{K}(0,\alpha )=\mathcal{K}\left(\alpha \right)$, $d(0,\alpha )=d\left(\alpha \right)$, …, in the case $s=0$. We can write (16) as follows

$${\mathcal{K}}_j\left(\alpha \right)=U_j^{-1}\left(\alpha \right){({\mathcal{K}}_{j-1}^{-1}\left(\alpha \right)+d_{j-1}\left(\alpha \right)I)}^{-1}{U}_j\left(\alpha \right)+2{\Theta }_j\left(\alpha \right),$$

(17)

where ${\Theta }_j\left(\alpha \right)=\cos {\varphi }_j{V}_j^{\ast }{N}_j{V}_j$. Note that all terms in the last formula are functions of $\alpha$, and are defined as in Section 2.5 with respect to $\alpha$.

5.1. Estimates of the Higher Derivative of Billiard Characteristics in ${\mathbb{R}}^n$

This section aims to demonstrate the differentiability of the billiard deformation characteristics in high dimensions with respect to $\alpha$. These characteristics are described in Section 5. Furthermore, we establish that the derivatives of these characteristics are bounded by constants independent of deformation parameter $\alpha \in [0,b]$ and the number of reflections $j\in {\mathbb{Z}}^{+}$. In particular, we show that the first and second derivatives are bounded, which holds significant relevance for the subsequent Section 7 and Section 8. The higher derivatives are bounded via induction. All corollaries, which are provided here, are based on Definition 1 and Theorem 5.

Corollary 1. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation with $r\ge 2,r^{\prime }\ge 1$, in ${\mathbb{R}}^n$ with $n\ge 3$. Let $q_j\left(\alpha \right)$ belong to $\partial K_{{\xi }_j}$. Then $q_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-1,r^{\prime }\}$, with respect to α, and there exist constants $C_q^{\left(s^{\prime }\right)}>0$ such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{q}_j\left(\alpha \right)}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_q^{\left(s^{\prime }\right)}.$$

Proof. 

Let $q_j\left(\alpha \right)={\phi }_j\left(u_j^{\left(1\right)}\left(\alpha \right),u_j^{\left(2\right)}\left(\alpha \right),\dots ,u_j^{(n-1)}\left(\alpha \right),\alpha \right)\in \partial K_{{\xi }_j}$. Then, from Definition 1 and Theorem 5, $q_j\left(\alpha \right)$ is ${\mathcal{C}}^{min\{r-1,r^{\prime }\}}$ with respect to $\alpha$. For the first derivatives of $q_j\left(\alpha \right)$ we have

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{q}_j}{\mathrm{d}\alpha }& =\sum _{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{\partial {\phi }_j}{\partial \alpha }.\hfill \end{array}$$

From Definition 1 and Theorem 5, there exists $C_q^{\left(1\right)}>0$ such that

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{q}_j}{\mathrm{d}\alpha }\parallel & \le (n-1)C_u^{\left(1\right)}+C_{\phi }^{(0,1)}=C_q^{\left(1\right)}.\hfill \end{array}$$

For the second derivative, we have

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{}^{order=2}{q}_j}{\mathrm{d}{\alpha }^{order=2}}& =\sum _{i=1}^{n-1}\sum _{k=1}^{n-1}\frac{\partial {}^2{\phi }_j}{\partial u_j^{\left(k\right)}\partial u_j^{\left(i\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\frac{\partial u_j^{\left(k\right)}}{\partial \alpha }+\sum _{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\frac{\partial {}^{order=2}{u}_j^{\left(i\right)}}{\partial {\alpha }^{order=2}}+2\sum _{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\alpha \frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{\partial {}^{order=2}{\phi }_j}{\partial {\alpha }^{order=2}}.\hfill \end{array}$$

As before, there exists $C_q^{\left(2\right)}>0$ such that

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{}^{order=2}{q}_j}{\mathrm{d}{\alpha }^{order=2}}\parallel & \le {(n-1)}^2{C}_{\phi }^{(2,0)}{\left(C_u^{\left(1\right)}\right)}^2+(n-1)C_u^{\left(2\right)}+2(n-1)C_{\phi }^{(1,1)}{C}_u^{\left(1\right)}+C_{\phi }^{(2,0)}=C_q^{\left(2\right)}.\hfill \end{array}$$

This constant is independent of j and $\alpha$. Continuing by induction, we can see that the $s^{\prime }$-th derivative of $q_j\left(\alpha \right)$ is bounded by a constant that depends only on $s^{\prime }$ and n. Thus, the statement is proved. □

Corollary 2. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 2,r^{\prime }\ge 1$. Then $d_j\left(\alpha \right)$ is ${\mathcal{C}}^s$, where $s=min\{r-1,r^{\prime }\}$, with respect to α, and there exist constants $C_v^{\left(s\right)}>0$ depending only on $s,$ and n such that

$$\left|\frac{\mathrm{d}{}^{order=s^{\prime }}{d}_j\left(\alpha \right)}{\mathrm{d}{\alpha }^{order=s^{\prime }}}\right|\le C_d^{\left(s^{\prime }\right)}.$$

Proof. 

Let $d_j=\parallel q_{j+1}\left(\alpha \right)-q_j\left(\alpha \right)\parallel$, where $q_j\left(\alpha \right)={\phi }_j\left(u_j^{\left(1\right)}\left(\alpha \right),u_j^{\left(2\right)}\left(\alpha \right),\dots ,u_j^{(n-1)}\left(\alpha \right),\alpha \right)$ belongs to $\partial K_{{\xi }_j}$. Then, from Corollary 1, $d_j\left(\alpha \right)$ is ${\mathcal{C}}^{min\{r-1,r^{\prime }\}}$ and its derivative with respect to $\alpha$ is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{d}_j}{\mathrm{d}\alpha }& =⟨\frac{{\phi }_{j+1}-{\phi }_j}{\parallel {\phi }_{j+1}-{\phi }_j\parallel },\frac{\partial {\phi }_{j+1}}{\partial \alpha }+\sum _{i=1}^{n-1}\frac{\partial {\phi }_{j+1}}{\partial u_{j+1}^{\left(i\right)}}\frac{\partial u_{j+1}^{\left(i\right)}}{\partial \alpha }-\frac{\partial {\phi }_j}{\partial \alpha }-\sum _{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }⟩.\hfill \end{array}$$

By using the estimations in Definition 1 and Theorem 5, we obtain

$$\begin{array}{cc}\hfill \left|\frac{\mathrm{d}{d}_j}{\mathrm{d}\alpha }\right|& \le 2C_{\phi }^{(0,1)}+2(n-1)C_u^{\left(1\right)}=C_d^{\left(1\right)},\hfill \end{array}$$

where $C_d^{\left(1\right)}>0$ is a constant depending only on n, and its second derivative is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{}^{order=2}{d}_j}{\mathrm{d}{\alpha }^{order=2}}& =\frac{{\left(\frac{\partial {\phi }_{j+1}}{\partial \alpha }+{\sum }_{i=1}^{n-1}\frac{\partial {\phi }_{j+1}}{\partial u_{j+1}^{\left(i\right)}}\frac{\partial u_{j+1}^{\left(i\right)}}{\partial \alpha }-\frac{\partial {\phi }_j}{\partial \alpha }-{\sum }_{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\right)}^2}{\parallel {\phi }_{j+1}-{\phi }_j\parallel }\hfill \\ \hfill & -\frac{{({\phi }_{j+1}-{\phi }_j)}^2{\left(\frac{\partial {\phi }_{j+1}}{\partial \alpha }+{\sum }_{i=1}^{n-1}\frac{\partial {\phi }_{j+1}}{\partial u_{j+1}^{\left(i\right)}}\frac{\partial u_{j+1}^{\left(i\right)}}{\partial \alpha }-\frac{\partial {\phi }_j}{\partial \alpha }-{\sum }_{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\right)}^2}{\parallel {\phi }_{j+1}-{\phi }_j{\parallel }^3}\hfill \\ \hfill & +\frac{{\phi }_{j+1}-{\phi }_j}{\parallel {\phi }_{j+1}-{\phi }_j\parallel }·(\frac{{\partial }^2{\phi }_{j+1}}{\partial {\alpha }^2}+\sum _{k=1}^{n-1}\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_{j+1}}{\partial u_{j+1}^{\left(k\right)}\partial u_{j+1}^{\left(i\right)}}\frac{\partial u_{j+1}^{\left(i\right)}}{\partial \alpha }\frac{\partial u_{j+1}^{\left(k\right)}}{\partial \alpha }\hfill \\ \hfill & +\sum _{i=1}^{n-1}\frac{\partial {\phi }_{j+1}}{\partial u_{j+1}^{\left(i\right)}}\frac{{\partial }^2{u}_{j+1}^{\left(i\right)}}{\partial {\alpha }^2}+\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_{j+1}}{\partial u_{j+1}^{\left(i\right)}\partial \alpha }\frac{\partial u_{j+1}^{\left(i\right)}}{\partial \alpha }-\frac{{\partial }^2{\phi }_j}{\partial {\alpha }^2}-\sum _{k=1}^{n-1}\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(k\right)}\partial u_j^{\left(i\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\frac{\partial u_j^{\left(k\right)}}{\partial \alpha }\hfill \\ \hfill & -\sum _{i=1}^{n-1}\frac{\partial {\phi }_j}{\partial u_j^{\left(i\right)}}\frac{{\partial }^2{u}_j^{\left(i\right)}}{\partial {\alpha }^2}-\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial \alpha }\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }).\hfill \end{array}$$

This is bounded by a constant $C_d^{\left(2\right)}$ depending only on n such that

$$\begin{array}{cc}\hfill \left|\frac{\mathrm{d}{}^{order=2}{d}_j}{\mathrm{d}{\alpha }^{order=2}}\right|& \le 2\frac{{\left(C_d^{\left(1\right)}\right)}^2}{d_{min}}+2C_{\phi }^{(0,2)}+2{(n-1)}^2{C}_{\phi }^{(2,0)}{\left(C_u^{\left(1\right)}\right)}^2+2(n-1)C_u^{\left(2\right)}+2(n-1)C_{\phi }^{(1,1)}{C}_u^{\left(1\right)}=C_d^{\left(2\right)}.\hfill \end{array}$$

Continuing by induction, we can see that the $s^{\prime }$-th derivative of $d_j\left(\alpha \right)$ is bounded by a constant which depends only on n and $s^{\prime }$. This proves the statement. □

Corollary 3. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 2,r^{\prime }\ge 1$. Let $v_j\left(\alpha \right)$ be the unit speed vector from $q_j\left(\alpha \right)$ to $q_{j+1}\left(\alpha \right)$. Then, $v_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-1,r^{\prime }\}$, with respect to α, and there exist constants $C_v^{\left(s^{\prime }\right)}>0$ depending only on n and $s^{\prime }$, such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{v}_j\left(\alpha \right)}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_v^{\left(s^{\prime }\right)}.$$

Proof. 

We can write $v_j\left(\alpha \right)=\frac{q_{j+1}\left(\alpha \right)-q_j\left(\alpha \right)}{d_j\left(\alpha \right)}$. And then by using Corollaries 1 and 2, the statement is proved. □

Corollary 4. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 3,r^{\prime }\ge 1,n\ge 3$. Let ${\nu }_j\left(\alpha \right)$ be the normal vector field to the boundary of obstacle $K_{{\xi }_j}$ at the point $q_j\left(\alpha \right)$. Then, ${\nu }_j\left(\alpha \right)$ is at least ${\mathcal{C}}^{s^{\prime }}$, where $s=min\{r-2,r^{\prime }\}$, and there exist constants $C_{\nu }^{\left(s^{\prime }\right)}>0$ depending only on n and $s^{\prime }$ such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{\nu }_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_{\nu }^{\left(s^{\prime }\right)}.$$

Proof. 

Let ${\nu }_j\left(\alpha \right)$ the normal vector field to the boundary of obstacle $K_{{\xi }_j}$ at the point $q_j\left(\alpha \right)$. We can write

$${\nu }_j\left(\alpha \right)=\frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\times \frac{\partial {\phi }_j}{\partial u_j^{\left(2\right)}}\times \cdots \times \frac{\partial {\phi }_j}{\partial u_j^{(n-1)}}.$$

Then, from Definition 1 and Theorem 5, ${\nu }_j$ is at least ${\mathcal{C}}^{min\{r-2,r^{\prime }\}}$ with respect to $\alpha$. To show its derivatives are bounded, we have

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\nu }_j}{\mathrm{d}\alpha }& =\left[\left(\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j}{\partial u^{\left(1\right)}\partial \alpha }\right)\times \frac{\partial {\phi }_j}{\partial u_j^{\left(2\right)}}\times \dots \times \frac{\partial {\phi }_j}{\partial u_j^{n-1}}\right]\hfill \\ \hfill & +\dots +\left[\frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\times \dots \times \left(\sum _{i=1}^{(n-1)}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{(n-1)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j}{\partial u_j^{(n-1)}\partial \alpha }\right)\right].\hfill \end{array}$$

Then, there exists a constant $C_{\nu }^{\left(1\right)}$ depending only on n such that

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{\nu }_j}{\mathrm{d}\alpha }\parallel & \le (n-1)\left[(n-1)C_{\phi }^{(2,0)}{C}_u^{\left(1\right)}+C_{\phi }^{(1,1)}\right]=C_{\nu }^{\left(1\right)}.\hfill \end{array}$$

(18)

Next, we want to show that the second derivative of ${\nu }_j\left(\alpha \right)$ is bounded by a constant.

To simplify the last derivative, we can write $\frac{\mathrm{d}{\nu }_j}{\mathrm{d}\alpha }={\Psi }_1+{\Psi }_2+\dots +{\Psi }_{n-1}$, where ${\Psi }_i$ corresponds to one square bracket […]. The first derivative of ${\Psi }_1$ with respect to $\alpha$ is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\Psi }_1}{\mathrm{d}\alpha }& =\left[\right(\sum _{k=1}^{n-1}\sum _{i=1}^{n-1}\frac{{\partial }^3{\phi }_j}{\partial u_j^{\left(k\right)}\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\frac{\partial u_j^{\left(k\right)}}{\partial \alpha }+\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\frac{{\partial }^2{u}_j^{\left(i\right)}}{\partial {\alpha }^2}\hfill \\ \hfill & +2\sum _{i=1}^{n-1}\frac{{\partial }^3{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}\partial \alpha }\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^3{\phi }_j}{\partial u_j^{\left(1\right)}\partial {\alpha }^2})\times \dots \times \frac{\partial {\phi }_j}{\partial u_j^{(n-1)}}]\hfill \\ \hfill & +\left[\left(\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j}{\partial u^{\left(1\right)}\partial \alpha }\right)\times \left(\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{\left(2\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j}{\partial u^{\left(1\right)}\partial \alpha }\right)\times \dots \times \frac{\partial {\phi }_j}{\partial u_j^{(n-1)}}\right]\hfill \\ \hfill & +\dots +\left[\left(\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j}{\partial u^{\left(1\right)}\partial \alpha }\right)\times \frac{\partial {\phi }_j}{\partial u_j^{\left(2\right)}}\times \dots \times \left(\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j}{\partial u_j^{\left(i\right)}\partial u_j^{(n-1)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j}{\partial u^{(n-1)}\partial \alpha }\right)\right].\hfill \end{array}$$

And

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{\Psi }_1}{\mathrm{d}\alpha }\parallel & \le {(n-1)}^2{C}_{\phi }^{(3,0)}{\left(C_u^{\left(1\right)}\right)}^2+(n-1)C_{\phi }^{(2,0)}{C}_u^{\left(2\right)}+2(n-1)C_{\phi }^{(2,1)}{C}_u^{\left(1\right)}+C_{\phi }^{(1,2)}\hfill \\ \hfill & +(n-1){\left[(n-1)C_{\phi }^{(2,0)}{C}_u^{\left(1\right)}+C_{\phi }^{(1,1)}\right]}^2=C_{\Psi }.\hfill \end{array}$$

We can obtain similar estimates for ${\displaystyle \parallel }\frac{\mathrm{d}{\Psi }_2}{\mathrm{d}\alpha }{\displaystyle \parallel },\dots ,{\displaystyle \parallel }\frac{\mathrm{d}{\Psi }_{n-1}}{\mathrm{d}\alpha }{\displaystyle \parallel }$. Therefore, there exists a constant $C_{\nu }^{\left(2\right)}>0$ depends only on n such that

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{}^{order=2}{\nu }_j}{\mathrm{d}{\alpha }^{order=2}}\parallel & \le (n-1)C_{\Psi }=C_{\nu }^{\left(2\right)}.\hfill \end{array}$$

So, we can see by induction that the $s^{\prime }$-th derivative, where $s^{\prime }=min\{r-2,r^{\prime }\}$, is bounded by a constant $C_{\nu }^{\left(s^{\prime }\right)}>0$, which depends only on n and $s^{\prime }$. This proves the statement. □

Corollary 5. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 3,r^{\prime }\ge 1,n\ge 3$. Let ${\varphi }_j\left(\alpha \right)$ the collision angle at $q_j\left(\alpha \right)$. Then, $\cos {\varphi }_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-2,r^{\prime }\}$, and there exist constants $C_{\varphi }^{\left(s^{\prime }\right)}>0$ depending only on n and $s^{\prime }$ such that

$$\left|\frac{\mathrm{d}{}^{order=s^{\prime }}\cos {\varphi }_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}\right|\le C_{\varphi }^{\left(s^{\prime }\right)}.$$

Proof. 

Recall that $\cos {\varphi }_j\left(\alpha \right)=⟨v_j\left(\alpha \right),{\nu }_j\left(\alpha \right)⟩$. Then, by using Corollaries 3 and 4, the statement is proved. □

Corollary 6. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1,n\ge 3$. Let $N_j\left(\alpha \right)$ be the shape operator of $\partial K_{{\xi }_j}$ at the point $q_j\left(\alpha \right)$. Then $N_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-3,r^{\prime }\}$, and there exist constants $C_{\kappa }^{\left(s^{\prime }\right)}>0$, which depend only on $s^{\prime }$ and n such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{N}_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_N^{\left(s^{\prime }\right)}.$$

Proof. 

The shape operator $N_j:T(\partial K_{{\xi }_j}\left(\alpha \right))\to T(\partial K_{{\xi }_j}\left(\alpha \right))$ at the point $q_j\left(\alpha \right)$, is defined by $N_j=\nabla {\nu }_j$, where ${\nu }_j$ is the normal vector field to the boundary of obstacle $K_{{\xi }_j}$ at the point $q_j\left(\alpha \right)$. First, from the expression of ${\nu }_j$, we have

$$\begin{array}{cc}\hfill \nabla {\nu }_j& =\nabla \left(\frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\times \frac{\partial {\phi }_j}{\partial u_j^{\left(2\right)}}\times \dots \times \frac{\partial {\phi }_j}{\partial u_j^{(n-1)}}\right).\hfill \\ \hfill & =\left[\left(\nabla \frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\right)\times \frac{\partial {\phi }_j}{\partial u_j^{\left(2\right)}}\times \dots \times \frac{\partial {\phi }_j}{\partial u_j^{(n-1)}}\right]+\dots .+\left[\frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\times \frac{\partial {\phi }_j}{\partial u_j^{\left(2\right)}}\times \dots \times \left(\nabla \frac{\partial {\phi }_j}{\partial u_j^{(n-1)}}\right)\right].\hfill \end{array}$$

(19)

And

$$\begin{array}{cc}\hfill \nabla \frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}& =\left(\sum _{i=1}^{n-1}\frac{\partial {}^2{\phi }_j^{\left(1\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}},\dots ,\sum _{i=1}^{n-1}\frac{\partial {}^2{\phi }_j^{\left(n\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\right)={\left(\sum _{i=1}^{n-1}\frac{\partial {}^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\right)}_{k=1}^n.\hfill \end{array}$$

This is bounded by a constant $C_{{\nabla }_1}^{\left(1\right)}$ depending only on n such that

$$\begin{array}{cc}\hfill \parallel \nabla \frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\parallel & \le n(n-1)C_{\phi }^{(2,0)}=C_{{\nabla }_1}.\hfill \end{array}$$

(20)

Also, $\nabla \frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}$ is ${\mathcal{C}}^{min\{r-3,r^{\prime }\}}$ with respect to $\alpha$ and its first derivative is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}\alpha }\nabla \frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}& =\frac{\mathrm{d}}{\mathrm{d}\alpha }{\left(\sum _{i=1}^{n-1}\frac{\partial {}^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\right)}_{k=1}^{\left(n\right)}\hfill \\ \hfill & ={\left(\sum _{i=1}^{n-1}\sum _{p=1}^{n-1}\frac{{\partial }^3{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(p\right)}\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}}\frac{\partial u_j^{\left(p\right)}}{\partial \alpha }+\sum _{i=1}^{n-1}\frac{\partial {}^3{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(1\right)}\partial \alpha }\right)}_{k=1}^{\left(n\right)}.\hfill \end{array}$$

(21)

Thus, there exists a constant $C_{{\nabla }_1}^{\left(1\right)}$ depending only on n such that

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}}{\mathrm{d}\alpha }\nabla \frac{\partial {\phi }_j}{\partial u_j^{\left(1\right)}}\parallel & \le n\left[{(n-1)}^2{C}_{\phi }^{(3,0)}{C}_u^{\left(1\right)}+(n-1)C_{\phi }^{(2,1)}\right]=C_{{\nabla }_1}^{\left(1\right)}.\hfill \end{array}$$

(22)

And then, by deriving (19) with respect to $\alpha$ and using the estimations in (18), (20) and (22), we obtain

$$\begin{array}{c}\hfill {\displaystyle \parallel }\frac{\mathrm{d}{N}_j}{\mathrm{d}\alpha }{\displaystyle \parallel }={\displaystyle \parallel }\frac{\mathrm{d}}{\mathrm{d}\alpha }\nabla {\nu }_j{\displaystyle \parallel }\le (n-1)\left[C_{{\nabla }_1}+C_{{\nabla }_1}^{\left(1\right)}{C}_{\nu }^{\left(1\right)}\right]=C_N^{\left(1\right)},\end{array}$$

(23)

which is a constant depending only on n. By induction, we can see that the $s^{\prime }$-derivative of $N_j\left(\alpha \right)$ with respect to $\alpha$ is bounded by a constant $C_N^{\left(s^{\prime }\right)}>0$ depending only on n and $s^{\prime }$. □

In the next Corollary, we want to show that

$$\begin{array}{c}\hfill {\tilde{u}}_j={\left(\frac{\partial {\phi }_j^{\left(k\right)}}{\partial u_j^{\left(t\right)}}\right)}_{k=1}^n\in T_{q_j}(\partial K)\end{array}$$

(24)

is differentiable with respect to $\alpha$ and its derivative is bounded by a constant independent of j and $\alpha$.

Corollary 7. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1$. For all $j=1,2,\dots m$, let ${\tilde{u}}_j\left(\alpha \right)$ as in (24). Then ${\tilde{u}}_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-2,r^{\prime }\}$, and there exist constants $C_{\tilde{u}}^{\left(s^{\prime }\right)}>0$, which are independent of j and α such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{\tilde{u}}_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_{\tilde{u}}^{\left(s^{\prime }\right)}.$$

Proof. 

Let ${\tilde{u}}_j\left(\alpha \right)$ be as in (24). From Definition 1 and Theorem 5, $\tilde{u}$ is ${\mathcal{C}}^{min\{r-2,r^{\prime }\}}$ with respect to $\alpha$ and its first derivative is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\tilde{u}}_j}{\mathrm{d}\alpha }& =\frac{\mathrm{d}}{\mathrm{d}\alpha }{\left(\frac{\partial {\phi }_j^{\left(k\right)}}{\partial u_j^{\left(t\right)}}\right)}_{k=1}^n={\left(\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(t\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(t\right)}\partial \alpha }\right)}_{k=1}^n.\hfill \end{array}$$

(25)

This is bounded by a constant $C_{\tilde{u}}^{\left(1\right)}>0$ as follows

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{\tilde{u}}_j\left(\alpha \right)}{\mathrm{d}\alpha }\parallel & \le n\left[(n-1)C_{\phi }^{(2,0)}{C}_u^{\left(1\right)}+C_{\phi }^{(1,1)}\right]=C_{\tilde{u}}^{\left(1\right)}.\hfill \end{array}$$

(26)

It is clear that this constant depends only on n. And the second derivative of ${\tilde{u}}_j\left(\alpha \right)$ with respect to $\alpha$ is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{}^{order=2}{\tilde{u}}_j}{\mathrm{d}{\alpha }^{order=2}}& =(\sum _{p=1}^{n-1}\sum _{i=1}^{n-1}\frac{{\partial }^3{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(p\right)}\partial u_j^{\left(i\right)}\partial u_j^{\left(t\right)}}\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\frac{\partial u_j^{\left(p\right)}}{\partial \alpha }+\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(t\right)}\partial \alpha }\frac{\partial u_j^{\left(i\right)}}{\partial \alpha }\hfill \\ \hfill & +\sum _{i=1}^{n-1}\frac{{\partial }^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(i\right)}\partial u_j^{\left(t\right)}}\frac{{\partial }^2{u}_j^{\left(i\right)}}{\partial {\alpha }^2}\sum _{p=1}^{n-1}\frac{{\partial }^3{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(p\right)}\partial u_j^{\left(t\right)}\partial \alpha }\frac{\partial u_j^{\left(p\right)}}{\partial \alpha }+\frac{{\partial }^2{\phi }_j^{\left(k\right)}}{\partial u_j^{\left(t\right)}\partial \alpha }{)}_{k=1}^n.\hfill \end{array}$$

(27)

Again from Definition 1 and Theorem 5, there exists a constant $C_{\tilde{u}}^{\left(2\right)}>0$ such that

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{}^{order=2}{\tilde{u}}_j\left(\alpha \right)}{\mathrm{d}{\alpha }^{order=2}}\parallel & \le n(n-1)\left[(n-1)C_{\phi }^{(3,0)}{\left(C_u^{\left(1\right)}\right)}^2+2C_{\phi }^{(2,1)}{C}_u^{\left(1\right)}+C_{\phi }^{(2,0)}{C}_u^{\left(2\right)}+C_{\phi }^{(1,2)}\right]=C_{\tilde{u}}^{\left(2\right)}.\hfill \end{array}$$

(28)

This constant depends only on n. By induction, we can see that the $s^{\prime }$-th derivative of ${\tilde{u}}_j$ with respect to $\alpha$ is bounded by a constant that depends only on $s^{\prime }$ and n. This proves the statement. □

The next Corollary follows from Corollaries 6 and 7.

Corollary 8. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1$. Let ${\kappa }_j\left(\alpha \right)$ be the normal curvature of the boundary $\partial K_{{\xi }_j}$ at the point $q_j\left(\alpha \right)$ in the direction $u_j\in T_{q_j}(\partial K_j)$, which is given by ${\kappa }_j\left(\alpha \right)=⟨N_j\left({\tilde{u}}_j\right),\widetilde{u_j}⟩$. Then ${\kappa }_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-3,r^{\prime }\}$ and there exist constants $C_{\kappa }^{\left(s^{\prime }\right)}>0$ which depend only on $s^{\prime }$ and n such that

$$\left|\frac{\mathrm{d}{}^{order=s^{\prime }}{\kappa }_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}\right|\le C_{\kappa }^{\left(s^{\prime }\right)}.$$

Corollary 9. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 3,r^{\prime }\ge 1$. Let $U_j$ and $U_j^{-1}$ be as in (6) and (7). Then, $U_j$ and $U_j^{-1}$ are at least ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-2,r^{\prime }\}$ and there exist constants $C_U^{\left(s^{\prime }\right)}>0$ depending only on n and $s^{\prime }$ such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{U}_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_n^{\left(s^{\prime }\right)}and{\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{U}_j^{-1}}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_n^{\left(s^{\prime }\right)}.$$

Proof. 

From (6) and (7), we can write

$$U_j=I-\frac{{\nu }_j{v}_{j-1}^{⊺}}{\cos {\varphi }_j}\mathrm{and}{U}_j^{-1}=I-\frac{{\nu }_j{v}_{j+1}^{⊺}}{\cos {\varphi }_j},$$

where the operators $U_j^{-1}$ and $U_j$ depend on $\alpha$. From Corollaries 3–5, $U_j$ and $U_j^{-1}$ are ${\mathcal{C}}^{min\{r-2,r^{\prime }\}}$. Moreover, we can see that their first derivatives are bounded by the same constant $C_n^{\left(1\right)}$ such that,

$$C_n^{\left(1\right)}=\frac{C_{\nu }^{\left(1\right)}+C_v^{\left(1\right)}}{\cos {\varphi }_{\max }},$$

(29)

where $\cos {\varphi }_{\max }\in [0,\pi /2)$. $C_n^{\left(1\right)}$ is only depending on n. And the second derivatives of $U_j$ and $U_j^{-1}$ with respect to $\alpha$ are bounded by $C_n^{\left(2\right)}$ such that

$$C_n^{\left(2\right)}=\frac{1}{{\left(\cos {\varphi }_{\max }\right)}^4}{C}_{\varphi }^{\left(1\right)}\left(4C_{\nu }^{\left(1\right)}+4C_v^{\left(1\right)}+C_{\varphi }^{\left(1\right)}\right)+C_{\nu }^{\left(2\right)}+C_{\nu }^{\left(1\right)}{C}_v^{\left(1\right)}+C_v^{\left(2\right)}+C_{\varphi }^{\left(2\right)}.$$

(30)

This constant depends only on n. By induction, we can see that the $s^{\prime }$-th derivatives of $U_j$ and $U_j^{-1}$ are bounded by a constant $C_n^{\left(s^{\prime }\right)}>0$ that depends only on n and $s^{\prime }$. This proves the statement. □

Corollary 10. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 3,r^{\prime }\ge 1$. Let $V_j^{\ast }$ and $V_j$ be as in (8) and (9). Then, $V_j^{\ast }$ and $V_j$ are at least ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-2,r^{\prime }\}$ and there exist constants $C_n^{\left(s^{\prime }\right)}>0$ depending only on n and $s^{\prime }$ such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{V}_j^{\ast }}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_n^{\left(s^{\prime }\right)}and{\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{V}_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_n^{\left(s^{\prime }\right)}.$$

Proof. 

From (8) and (9), we can write

$$V=I-\frac{v_j{\nu }_j^{⊺}}{\cos {\varphi }_j}\mathrm{and}{V}^{\ast }=I-\frac{{\nu }_j{v}_j^{⊺}}{\cos {\varphi }_j}.$$

Then, the rest of the proof is the same as the proof of Corollary 9. □

The next Corollary follows from Corollaries 5, 6 and 10.

Corollary 11. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1$. Let ${\Theta }_j\left(\alpha \right)=\cos {\varphi }_j{V}_j^{\ast }{N}_j{V}_j$. Then, ${\Theta }_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-3,r^{\prime }\}$ and there exist constants $C_{\Theta }^{\left(s^{\prime }\right)}>0$ depending only on $s^{\prime }$ and n such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{\Theta }_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_{\Theta }^{\left(s^{\prime }\right)}.$$

In the next Corollary, we want to show that ${\widehat{w}}_j\left(\alpha \right)\in T_{q_j}\left(X_j\left(\alpha \right)\right)$, such that

${\widehat{w}}_j=w_j/\parallel w_j\parallel$ and

$$w_j={\tilde{u}}_j-\frac{⟨{\tilde{u}}_j,v_j⟩}{\cos {\varphi }_j}{\nu }_j,$$

(31)

where ${\tilde{u}}_j$ is as in (24).

Corollary 12. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1,n\ge 3$. For all $j=1,2,\dots m$, let $w_j\left(\alpha \right)$ be as in (31). Then, ${\widehat{w}}_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-2,r^{\prime }\}$, and there exist constants $C_{\widehat{w}}^{\left(s^{\prime }\right)}>0$, which are independent of j and α such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{\widehat{w}}_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_{\widehat{w}}^{\left(s^{\prime }\right)}.$$

Proof. 

Let $w_j\left(\alpha \right)\in {\mathcal{J}}_j$ as in (31), where ${\mathcal{J}}_j$ is the hyperplane to the convex front ${\tilde{X}}_j$ at the point $q_j\left(\alpha \right)$. From Corollaries 3, 4 and 7, ${\widehat{w}}_j$ is at least ${\mathcal{C}}^{min\{r-2,r^{\prime }\}}$ with respect to $\alpha$, and there exists a constant $C_{\widehat{w}}^{\left(1\right)}>0$ such that

$${\displaystyle \parallel }\frac{\mathrm{d}{\widehat{w}}_j}{\mathrm{d}\alpha }{\displaystyle \parallel }\le \frac{2}{(1+\cos {\varphi }^{+}){\cos }^2{\varphi }^{+}}\left(2C_{\tilde{u}}^{\left(1\right)}+C_v^{\left(1\right)}+C_{\nu }^{\left(1\right)}+C_{\varphi }^{\left(1\right)}\right)=C_{\widehat{w}}^{\left(1\right)}.$$

This constant depends only on n. By induction, we can see that the $s^{\prime }$-th derivative of ${\widehat{w}}_j$ with respect to $\alpha$ is bounded by a constant that depends only on $s^{\prime }$ and n. This proves the statement. □

Corollary 13. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1$. Then for all $j=1,2,\dots ,m$, ${\mathcal{K}}_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s=min\{r-3,r^{\prime }\}$ and there exists a constant $C_{\mathcal{K}}^{\left(s^{\prime }\right)}>0$, which depends on $s^{\prime }$ and n such that

$${\displaystyle \parallel }\frac{\mathrm{d}{}^{order=s^{\prime }}{\mathcal{K}}_j\left(\alpha \right)}{\mathrm{d}{\alpha }^{order=s^{\prime }}}{\displaystyle \parallel }\le C_{\mathcal{K}}^{\left(s^{\prime }\right)}.$$

Proof. 

From (16), recall that

$${\mathcal{K}}_j\left(\alpha \right)=U_j^{-1}{\mathcal{K}}_j^{-}{U}_j+2{\Theta }_j,$$

where ${\mathcal{K}}_j^{-}\left(\alpha \right)={\mathcal{K}}_{j-1}{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}$. Given that every term in the right-hand side of the equation is differentiable with respect to $\alpha$, and we can derive ${\mathcal{K}}_{j-1}$ in terms of ${\mathcal{K}}_0$ on the right-hand side, where ${\mathcal{K}}_0$ is independent of $\alpha$, we can conclude that ${\mathcal{K}}_j$ is differentiable with respect to $\alpha$. Specifically, ${\mathcal{K}}_j$ is at least ${\mathcal{C}}^{min\{r-3,r^{\prime }\}}$, where $r\ge 4$ and $r^{\prime }\ge 1$. This result follows from the Corollaries 11 and 9.

Next, we want to show that the first derivative of ${\mathcal{K}}_j$ with respect to $\alpha$ is bounded by a constant independent of $\alpha$ and j. First, we have

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\mathcal{K}}_j^{-}}{\mathrm{d}\alpha }& =\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}-{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}(d_{j-1}\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }+\frac{\mathrm{d}{d}_{j-1}}{\mathrm{d}\alpha }{\mathcal{K}}_{j-1}){(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\hfill \\ \hfill & =\left[\left(I-d_{j-1}{\mathcal{K}}_{j-1}{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\right)\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }\right]{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\hfill \\ \hfill & -{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\frac{\mathrm{d}{d}_{j-1}}{\mathrm{d}\alpha }{\mathcal{K}}_{j-1}^2{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\hfill \\ \hfill & ={(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}-{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\frac{\mathrm{d}{d}_{j-1}}{\mathrm{d}\alpha }{\mathcal{K}}_{j-1}^2{(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}\hfill \\ \hfill & ={\mathcal{D}}_{j-1}\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }{\mathcal{D}}_{j-1}+{\mathcal{E}}_{j-1}\hfill \end{array}$$

(32)

where ${\mathcal{D}}_{j-1}\left(\alpha \right)={(I+d_{j-1}{\mathcal{K}}_{j-1})}^{-1}$ and ${\mathcal{E}}_{j-1}=-\frac{\mathrm{d}{d}_{j-1}}{\mathrm{d}\alpha }{\mathcal{D}}_{j-1}{\mathcal{K}}_{j-1}^2{\mathcal{D}}_{j-1}$. Second, we derive

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\mathcal{K}}_j}{\mathrm{d}\alpha }& =U_j^{-1}\frac{\mathrm{d}{\mathcal{K}}_{j-1}^{-}}{\mathrm{d}\alpha }{U}_j-U_j^{-1}\frac{\mathrm{d}{U}_j^{-1}}{\mathrm{d}\alpha }{U}_j^{-1}{\mathcal{K}}_{j-1}{U}_j+U_j^{-1}{\mathcal{K}}_{j-1}^{-}\frac{\mathrm{d}{U}_j}{\mathrm{d}\alpha }+2\frac{\mathrm{d}{\Theta }_j}{\mathrm{d}\alpha }.\hfill \end{array}$$

From (32), we obtain

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\mathcal{K}}_j}{\mathrm{d}\alpha }& =U_j^{-1}{\mathcal{D}}_{j-1}\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }{\mathcal{D}}_{j-1}{U}_j+U_j^{-1}{\mathcal{E}}_{j-1}{U}_j-\frac{\mathrm{d}{U}_{j-1}^{-1}}{\mathrm{d}\alpha }{\mathcal{K}}_{j-1}+{\mathcal{K}}_{j-1}^{-}\frac{\mathrm{d}{U}_{j-1}}{\mathrm{d}\alpha }+2\frac{\mathrm{d}{\Theta }_j}{\mathrm{d}\alpha }.\hfill \end{array}$$

Let ${\tilde{\mathcal{P}}}_j=U_j^{-1}{\mathcal{D}}_{j-1}$, ${\mathcal{P}}_j={\mathcal{D}}_{j-1}{U}_j$, and ${\mathcal{R}}_j=U_j^{-1}{\mathcal{E}}_{j-1}{U}_j-\frac{\mathrm{d}{U}_j^{-1}}{\mathrm{d}\alpha }{\mathcal{K}}_{j-1}+{\mathcal{K}}_{j-1}^{-}\frac{\mathrm{d}{U}_j}{\mathrm{d}\alpha }+2\frac{\mathrm{d}{\Theta }_j}{\mathrm{d}\alpha }$. So, we can write

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\mathcal{K}}_j}{\mathrm{d}\alpha }& ={\tilde{\mathcal{P}}}_j\frac{\mathrm{d}{\mathcal{K}}_{j-1}}{\mathrm{d}\alpha }{\mathcal{P}}_j+{\mathcal{R}}_j.\hfill \end{array}$$

To obtain the estimation, we have

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\mathcal{K}}_j}{\mathrm{d}\alpha }& ={\tilde{\mathcal{P}}}_j\left[{\tilde{\mathcal{P}}}_{j-1}\frac{\mathrm{d}{\mathcal{K}}_{j-2}}{\mathrm{d}\alpha }{\mathcal{P}}_{j-1}+{\mathcal{R}}_{j-1}\right]{\mathcal{P}}_j+{\mathcal{R}}_j\hfill \\ \hfill & ={\tilde{\mathcal{P}}}_j\left[{\tilde{\mathcal{P}}}_{j-1}\left({\tilde{\mathcal{P}}}_{j-2}\frac{\mathrm{d}{\mathcal{K}}_{j-3}}{\mathrm{d}\alpha }{\mathcal{P}}_{j-2}+{\mathcal{R}}_{j-2}\right){\mathcal{P}}_j+{\mathcal{R}}_{j-1}\right]{\mathcal{P}}_{j-1}+{\mathcal{R}}_j\hfill \\ \hfill & ={\tilde{\mathcal{P}}}_j{\tilde{\mathcal{P}}}_{j-1}\dots {\tilde{\mathcal{P}}}_1\frac{\mathrm{d}{\mathcal{K}}_0}{\mathrm{d}\alpha }{\mathcal{P}}_1\dots {\mathcal{P}}_{j-1}{\mathcal{P}}_j+{\tilde{\mathcal{P}}}_j{\mathcal{R}}_{j-1}{\mathcal{P}}_j+\dots +{\tilde{\mathcal{P}}}_j{\tilde{\mathcal{P}}}_{j-1}\dots {\tilde{\mathcal{P}}}_2{\mathcal{R}}_1{\mathcal{P}}_2\dots {\mathcal{P}}_{j-1}{\mathcal{P}}_j+{\mathcal{R}}_j.\hfill \end{array}$$

Since ${\mathcal{K}}_0$ is independent of $\alpha$, then

$$\begin{array}{c}\hfill \frac{\mathrm{d}{\mathcal{K}}_j}{\mathrm{d}\alpha }={\tilde{\mathcal{P}}}_j{\mathcal{R}}_{j-1}{\mathcal{P}}_j+\dots +{\tilde{\mathcal{P}}}_j\dots {\tilde{\mathcal{P}}}_2{\mathcal{R}}_1{\mathcal{P}}_2\dots {\mathcal{P}}_j+{\mathcal{R}}_j.\end{array}$$

From (10) and Corollaries 2, 9 and 11, we have ${\displaystyle \parallel }{\mathcal{P}}_j{\displaystyle \parallel }\le {\mathcal{D}}_{\max }$ and ${\displaystyle \parallel }{\tilde{\mathcal{P}}}_j{\displaystyle \parallel }\le {\mathcal{D}}_{\max }$, where ${\mathcal{D}}_{\max }=\frac{1}{1+d_{min}{\mu }_{min}}$, and

$${\displaystyle \parallel }{\mathcal{R}}_j{\displaystyle \parallel }\le {\mathcal{R}}_{\max }={\left({\mathcal{D}}_{\max }\right)}^2{C}_d^{\left(1\right)}{\left({\eta }_{\max }\right)}^2+C_w^{\left(1\right)}{\eta }_{\max }+\frac{C_w^{\left(1\right)}}{d_{min}}+2C_{\Theta }^{\left(1\right)}.$$

And then

$$\begin{array}{cc}\hfill \parallel \frac{\mathrm{d}{\mathcal{K}}_j^{}}{\mathrm{d}\alpha }\parallel & \le \left[1+{\left({\mathcal{D}}_{\max }\right)}^2+{\left({\mathcal{D}}_{\max }\right)}^4+\dots +{\left({\mathcal{D}}_{\max }\right)}^{2(j-1)}\right]{\mathcal{R}}_{\max }\hfill \\ \hfill & =\frac{(1-{\left({\mathcal{D}}_{\max }\right)}^{2j}}{1-{\left({\mathcal{D}}_{\max }\right)}^2}{\mathcal{R}}_{\max }.\hfill \end{array}$$

For a large j, ${\left({\mathcal{D}}_{\max }\right)}^{2j}$ converges to 0. Thus, there exists a constant $C_{{\mathcal{K}}^{}}^{\left(1\right)}>0$ independent of j and $\alpha$, such that

$${\displaystyle \parallel }\frac{\mathrm{d}{\mathcal{K}}_j^{}}{\mathrm{d}\alpha }{\displaystyle \parallel }\le C_{{\mathcal{K}}^{}}^{\left(1\right)}.$$

By using the same approach we can see that the $s^{\prime }$-th derivative of ${\mathcal{K}}_j$ is bounded by a constant $C_{{\mathcal{K}}^{}}^{\left(s^{\prime }\right)}>0$, which depends only on n and $s^{\prime }$. □

The next Corollary follows from Corollaries 12 and 13.

Corollary 14. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1$, and $n\ge 3$. Let $k_j\left(\alpha \right)$ be the normal curvature of the unstable manifold ${\tilde{X}}_j$ at the point $q_j\left(\alpha \right)$ in the direction ${\widehat{w}}_j\in T_{q_j}\left({\tilde{X}}_j\right)$, which is given by $k_j\left(\alpha \right)=⟨{\mathcal{K}}_j\left({\widehat{w}}_j\right),{\widehat{w}}_j⟩$. Then, $k_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-3,r^{\prime }\}$ and there exist constants $C_k^{\left(s^{\prime }\right)}>0$, which depend only on $s^{\prime }$ and n such that

$$\left|\frac{\mathrm{d}{}^{order=s^{\prime }}{k}_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}\right|\le C_k^{\left(s^{\prime }\right)}.$$

Corollary 15. 

Let $K\left(\alpha \right)$ be a ${\mathcal{C}}^{r,r^{\prime }}$ billiard deformation in ${\mathbb{R}}^n$ with $r\ge 4,r^{\prime }\ge 1$. Let ${\ell }_j\left(\alpha \right)$ be as defined in (14). Then, ${\ell }_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-3,r^{\prime }\}$, and there exist constants $C_{\ell }^{\left(s^{\prime }\right)}>0$, which depend on $s^{\prime }$ and n such that

$$\left|\frac{\mathrm{d}{}^{order=s^{\prime }}{\ell }_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}\right|\le C_{\ell }^{\left(s^{\prime }\right)}.$$

Proof. 

Recall that

$${(1+d_j\left(\alpha \right){\ell }_j\left(\alpha \right))}^2={\parallel {\widehat{w}}_j\left(\alpha \right)+\left(d_j\left(\alpha \right)\right){\mathcal{K}}_j\left({\widehat{w}}_j\left(\alpha \right)\right)\parallel }^2.$$

From this and Corollaries 2, 12 and 13, ${\ell }_j\left(\alpha \right)$ is ${\mathcal{C}}^{s^{\prime }}$, where $s^{\prime }=min\{r-3,r^{\prime }\}$. And its first derivative with respect to $\alpha$ is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\ell }_j}{\mathrm{d}\alpha }& =\frac{1}{d_j}\left[\frac{{\widehat{w}}_j+d_j{\mathcal{K}}_j\left({\widehat{w}}_j\right)}{\parallel {\widehat{w}}_j+d_j{\mathcal{K}}_j\left({\widehat{w}}_j\right)\parallel }(\frac{\mathrm{d}}{\mathrm{d}\alpha }{\widehat{w}}_j+\frac{\mathrm{d}}{\mathrm{d}\alpha }{d}_j{\mathcal{K}}_j\left({\widehat{w}}_j\right)+d_j\frac{\mathrm{d}}{\mathrm{d}\alpha }{\mathcal{K}}_j\left({\widehat{w}}_j\right)+d_j\mathcal{K}\frac{\mathrm{d}}{\mathrm{d}\alpha }{\widehat{w}}_j-\frac{\mathrm{d}}{\mathrm{d}\alpha }{d}_j{\ell }_j\right].\hfill \end{array}$$

And again by Corollaries 2, 12 and 13, we have

$$\begin{array}{c}\hfill \left|\frac{\mathrm{d}{\ell }_j}{\mathrm{d}\alpha }\right|\le \frac{1}{d_{min}}\left(C_{w_1}^{\left(1\right)}+2C_d^{\left(1\right)}{\eta }_{\max }+d_{\max }{C}_{{\mathcal{K}}_1}^{\left(1\right)}\right).\end{array}$$

Then, there exists a $C_{{\ell }_1}^{\left(1\right)}>0$ depending on n, such that $\left|\frac{\mathrm{d}{\ell }_j}{\mathrm{d}\alpha }\right|\le C_{{\ell }_1}^{\left(1\right)}$. By induction, we can see that $\left|\frac{\mathrm{d}{}^{order=s^{\prime }}{\ell }_j}{\mathrm{d}{\alpha }^{order=s^{\prime }}}\right|\le C_{{\ell }_j}^{\left(s^{\prime }\right)}$, where $C_{{\ell }_j}^{\left(s^{\prime }\right)}$ is constant depending only on $s^{\prime }$ and n. □

6. Estimation the Largest Lyapunov Exponent for Open Billiard Deformation

Recall from Section 4 that $B_{\alpha }:M_{\alpha }⟶M_{\alpha }$ and $R_{\alpha }:M_{\alpha }⟶\Sigma$, for a small $\alpha \in [0,a]$ where $a\in {\mathbb{R}}^{+}$. According to Theorem 3 there exists a subset $A_{\alpha }$ of $\Sigma$ with $\mu \left(A_{\alpha }\right)=1$ such that

$${\lambda }_1\left(\alpha \right)=\underset{m\to \infty }{lim}\frac{1}{m}log\parallel D_x{B}_{\alpha }^m\left(w\right)\parallel$$

(33)

for $\mu$-almost $x\left(\alpha \right)\in M_{\alpha }$ with $R_{\alpha }\left(x\right)\in A_{\alpha }$, and all $w\in E_1^u\left(x\right)\oplus E_2^u\left(x\right)\oplus \cdots \oplus E_k^u\left(x\right)\setminus E_2^u\left(x\right)\oplus E_3^u\left(x\right)\oplus \cdots \oplus E_k^u\left(x\right)$. In case $\alpha =0$, we have $A_0$ with $\mu \left(A_0\right)=1$, which we showed in Section 3.

The next Lemma shows that for $\mu$-almost $x\left(\alpha \right)\in M_{\alpha }$, we can choose $\xi \in \Sigma$ such that the last Formula (33) holds for $\alpha ={\alpha }_p$ and $x\left({\alpha }_p\right)$ for all $p=1,2,\dots$, and also for $\alpha =0$ and $x\left(0\right)$, which refers to the initial open billiard.

Lemma 1. 

Given an arbitrary sequence

$${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_p,\dots$$

of elements of $[0,b]$, for μ-almost all $\xi \in \Sigma$ the formula (33) is valid for $\alpha ={\alpha }_p$ and $x=R_{\alpha }^{-1}\left(\xi \right)$ for all $p=1,2,\dots$ and also for $\alpha =0$ and $x=R^{-1}\left(\xi \right)$.

Proof. 

See [8]. □

From the formula for the largest Lyapunov exponent (11), we can write the Lyapunov exponents for $K\left(\alpha \right)$ and $K\left(0\right)$ as follows:

$$\begin{array}{cc}\hfill {\lambda }_1\left(\alpha \right)& =\underset{m\to \infty }{lim}\frac{1}{m}\sum _{j=1}^{m}log\left(1+d_j\left(\alpha \right){\ell }_j\left(\alpha \right)\right)=\underset{m\to \infty }{lim}{\lambda }_1^{\left(m\right)}\left(\alpha \right),\hfill \\ \hfill {\lambda }_1\left(0\right)& =\underset{m\to \infty }{lim}\frac{1}{m}\sum _{j=1}^{m}log\left(1+d_j\left(0\right){\ell }_j\left(0\right)\right)=\underset{m\to \infty }{lim}{\lambda }_1^{\left(m\right)}\left(0\right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\lambda }_1^{\left(m\right)}\left(\alpha \right)=\frac{1}{m}\sum _{j=1}^{m}log\left(1+d_j\left(\alpha \right){\ell }_j\left(\alpha \right)\right)\\ \hfill \mathrm{and}{\lambda }_1^{\left(m\right)}\left(0\right)=\frac{1}{m}\sum _{j=1}^{m}log\left(1+d_j\left(0\right){\ell }_j\left(0\right)\right).\end{array}$$

(34)

7. The Continuity of the Largest Lyapunov Exponent for Non-Planar Billiard Deformation

This section provides a rigorous proof that the largest Lyapunov exponent of the billiard deformation in ${\mathbb{R}}^n$, where $n\ge 3$, is a continuous function at every perturbation parameter $\alpha \in [0,b]$. Our proof closely follows the argument used in the proof of the continuity of the largest Lyapunov exponent for planar billiard deformation [8], as well as the results established in Section 5.1 and previous sections.

Here we prove Theorem 1:

Proof of Theorem 1. 

We will show that ${\lambda }_1\left(\alpha \right)$ is continuous at $\alpha =0$. From this, the continuity is followed at every $\alpha \in [0,b]$. To prove that, we will show that for any sequence of points ${\alpha }_p\in [0,b]$ as in Lemma 1 with ${\alpha }_p\to 0$ when $p\to \infty$, ${\lambda }_1^m\left({\alpha }_p\right)\to {\lambda }_1^m\left(0\right)$.

From (34), we have

$$\begin{array}{cc}\hfill {\lambda }_1^m\left({\alpha }_p\right)-{\lambda }_1^m\left(0\right)& =\frac{1}{m}\sum _{i=1}^m\left[log(1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right))-log(1+d_j\left(0\right){\ell }_j\left(0\right))\right].\hfill \end{array}$$

(35)

Let $f_j\left({\alpha }_p\right)=log(1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right))$. From Corollaries 2 and 15, $f_j\left({\alpha }_p\right)$ is ${\mathcal{C}}^2$ with respect to ${\alpha }_p$, and its first derivative is

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{f}_j\left({\alpha }_p\right)}{\mathrm{d}{\alpha }_p}& =\frac{\frac{\mathrm{d}{d}_j\left({\alpha }_p\right)}{\mathrm{d}{\alpha }_p}{\ell }_j\left({\alpha }_p\right)+d_j\left({\alpha }_p\right)\frac{\mathrm{d}{\ell }_j\left({\alpha }_p\right)}{\mathrm{d}{\alpha }_p}}{1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right)}.\hfill \end{array}$$

(36)

In addition, there exists a constant $C_f^{\left(1\right)}>0$ independent of j and ${\alpha }_p$ such that,

$$\begin{array}{cc}\hfill \left|\frac{\mathrm{d}{f}_j\left({\alpha }_p\right)}{\mathrm{d}{\alpha }_p}\right|& \le \frac{C_d^{\left(1\right)}{\eta }_{\max }+d_{\max }{C}_{\ell }^{\left(1\right)}}{1+d_{min}{\mu }_{min}}=C_f^{\left(1\right)}.\hfill \end{array}$$

(37)

From this and applying the Mean Value Theorem to (35), we obtain

$$\begin{array}{cc}\hfill {\displaystyle |}{\lambda }_1^m\left({\alpha }_p\right)-{\lambda }_1^m\left(0\right){\displaystyle |}& \le \frac{1}{m}\sum _{i=1}^m{\alpha }_p{C}_f^{\left(1\right)}={\alpha }_p{C}_f^{\left(1\right)}.\hfill \end{array}$$

By letting ${\alpha }_p$ approach 0 as p approaches infinity, and then letting m approach infinity, we obtain ${\lambda }_1\left({\alpha }_p\right)$ approaching ${\lambda }_1\left(0\right)$, for any sequence ${\alpha }_1>{\alpha }_2>\dots >{\alpha }_p>\dots \to 0$ as $p\to \infty$ in $[0,b]$ as in Lemma 1. This proves the statement. □

8. The Differentiability of the Largest Lyapunov Exponent for the Non-Planar Billiard Deformation

In this section, we present a proof of the differentiability of the largest Lyapunov exponent ${\lambda }_1$ for the billiard deformation in ${\mathbb{R}}^n$, with $n\ge 3$, with respect to a small perturbation $\alpha$. This proof is based on the findings in the preceding sections and resembles the proof in [8] for the case of planar billiard deformation.

Here is the proof of Theorem 2:

Proof of Theorem 2. 

We prove differentiability at $\alpha =0$, and this implies differentiability for any $\alpha \in [0,b]$. To prove the differentiability at $\alpha =0$, we have to show that there exists

$$\begin{array}{c}\hfill \underset{\alpha \to 0}{lim}\frac{{\lambda }_1\left(\alpha \right)-{\lambda }_1\left(0\right)}{\alpha }.\end{array}$$

Equivalently, there exists a number F such that

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}\frac{{\lambda }_1\left({\alpha }_p\right)-{\lambda }_1\left(0\right)}{{\alpha }_p}=F,\end{array}$$

for any sequence ${\alpha }_1>{\alpha }_2>\dots >{\alpha }_p>\dots \to 0$ as $p\to \infty$ in $[0,b]$ as in Lemma 1. By using this Lemma and the expressions of ${\lambda }_1^m\left(\alpha \right)$ for $\alpha ={\alpha }_p$ and ${\lambda }_1^m\left(0\right)$ for $\alpha =0$ in (34), we have ${\lambda }_1^{\left(m\right)}\left({\alpha }_p\right)\to {\lambda }_1\left({\alpha }_p\right)$ and ${\lambda }_1^{\left(m\right)}\left(0\right)\to {\lambda }_1\left(0\right)$ when $m\to \infty$. Let $f_j\left({\alpha }_p\right)=log\left(1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right)\right)$ and $f_j\left(0\right)=log\left(1+d_j\left(0\right){\ell }_j\left(0\right)\right)$. And Taylor–Lagrange formula gives

$$\begin{array}{cc}\hfill f_j\left({\alpha }_p\right)& =f_j\left(0\right)+{\alpha }_p{\dot{f}}_j\left(0\right)+\frac{{\alpha }_p^2}{2}{\ddot{f}}_j\left(r_j\left({\alpha }_p\right)\right)\hfill \end{array}$$

for some $r_j\left({\alpha }_p\right)\in [0,{\alpha }_p]$. Then,

$$\begin{array}{cc}\hfill \frac{f_j\left({\alpha }_p\right)-f_j\left(0\right)}{{\alpha }_p}-{\dot{f}}_j\left(0\right)& =\frac{{\alpha }_p}{2}{\ddot{f}}_j\left(r_j\left({\alpha }_p\right)\right).\hfill \end{array}$$

After multiplying by $m^{-1}$ and summing up for $j=1$ to m, we have

$$\begin{array}{c}\hfill \frac{1}{m}\sum _{j=1}^m\left[\frac{f_j\left({\alpha }_p\right)-f_j\left(0\right)}{{\alpha }_p}\right]-\frac{1}{m}\sum _{j=1}^m{\dot{f}}_j\left(0\right)=\frac{1}{m}\sum _{j=1}^m\left[\frac{{\alpha }_p}{2}{\ddot{f}}_j\left(r_j\left({\alpha }_p\right)\right)\right].\end{array}$$

Thus,

$$\begin{array}{cc}\hfill \frac{{\lambda }_1^{\left(m\right)}\left({\alpha }_p\right)-{\lambda }_1^{\left(m\right)}\left(0\right)}{{\alpha }_p}-F_m& =\frac{1}{m}\sum _{j=1}^m\left[\frac{{\alpha }_p}{2}{\ddot{f}}_j\left(r_j\left({\alpha }_p\right)\right)\right],\hfill \end{array}$$

where $F_m=\frac{1}{m}{\sum }_{j=1}^m{\dot{f}}_j\left(0\right)$. Next, we showed the first derivative of $f_j\left({\alpha }_p\right)$ in (36). As $f_j\left({\alpha }_p\right)$ is ${\mathcal{C}}^2$, we can now proceed to calculate the second derivative as follows:

$$\begin{array}{cc}\hfill {\ddot{f}}_j\left({\alpha }_p\right)& =\frac{\left({\ddot{d}}_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right)+2{\dot{d}}_j\left({\alpha }_p\right){\dot{\ell }}_j\left({\alpha }_p\right)+d_j\left({\alpha }_p\right){\ddot{\ell }}_j\left({\alpha }_p\right)\right)\left(1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right)\right)}{{\left(1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right)\right)}^2}\hfill \\ \hfill & -\frac{{\left({\dot{d}}_j\left({\alpha }_p\right)\ell \left({\alpha }_p\right)+d_j\left({\alpha }_p\right){\dot{\ell }}_j\left({\alpha }_p\right)\right)}^2}{{\left(1+d_j\left({\alpha }_p\right){\ell }_j\left({\alpha }_p\right)\right)}^2}.\hfill \end{array}$$

According to Corollaries 2 and 15, there exists a constant $C_f^{\left(2\right)}>0$ that does not depend on j and ${\alpha }_p$ such that,

$$\begin{array}{cc}\hfill \left|{\ddot{f}}_j\left({\alpha }_p\right)\right|& \le \frac{\left(C_d^{\left(2\right)}{\eta }_{\max }+2C_d^{\left(1\right)}{C}_{\ell }^{\left(1\right)}+d_{\max }{C}_{\ell }^{\left(2\right)}\right)\left(1+d_{\max }{\eta }_{\max }\right)}{{\left(1+d_{min}{\mu }_{min}\right)}^2}\hfill \\ \hfill & +\frac{{\left(C_d^{\left(1\right)}{\eta }_{\max }+d_{\max }{C}_{\ell }^{\left(1\right)}\right)}^2}{{\left(1+d_{min}{\mu }_{min}\right)}^2}=C_f^{\left(2\right)}.\hfill \end{array}$$

Therefore, ${\displaystyle |}{\ddot{f}}_j\left(r_j\left({\alpha }_p\right)\right){\displaystyle |}\le C_f^{\left(2\right)},$ implying that

$$\begin{array}{ccc}\hfill |\frac{{\lambda }_1^{\left(m\right)}\left({\alpha }_p\right)-{\lambda }_1^{\left(m\right)}\left(0\right)}{{\alpha }_p}-F_m|& \le \frac{1}{m}\sum _{j=1}^m\frac{{\alpha }_p}{2}\left|{\ddot{f}}_j\left(t_j\left({\alpha }_p\right)\right)\right|\hfill & \hfill \le \frac{C_f^{\left(2\right)}}{2}{\alpha }_p.\end{array}$$

From (37), we have $|F_m|\le \frac{1}{m}{\sum }_{j=1}^m|{\dot{f}}_j\left(0\right)|\le C_f^{\left(1\right)}$. So, by the Bolzano–Weierstrass Theorem, there exists a convergent subsequence $\left\{F_{m_h}\right\}$ of $\left\{F_m\right\}$ such that ${lim}_{h\to \infty }{F}_{m_h}=F$. Then

$$\begin{array}{cc}\hfill |\frac{{\lambda }_1^{\left(m_h\right)}\left({\alpha }_p\right)-{\lambda }_1^{\left(m_h\right)}\left(0\right)}{{\alpha }_p}-F_{m_h}|& \le \frac{C_f^{\left(2\right)}}{2}{\alpha }_p.\hfill \end{array}$$

By letting $h\to \infty$, and letting ${lim}_{p\to \infty }{\alpha }_p=0$, we obtain for every sequence ${\alpha }_1>{\alpha }_2>\dots >{\alpha }_p>\dots \to 0$ as $p\to \infty$ in $[0,b]$ as in Lemma 1,

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}\frac{{\lambda }_1\left({\alpha }_p\right)-{\lambda }_1\left(0\right)}{{\alpha }_p}=F.\end{array}$$

Thus, there exists $F={lim}_{m\to \infty }\frac{1}{m}{\sum }_{j=1}^m{\dot{f}}_j\left(0\right)$. This holds for any subsequence $\left\{m_h\right\}$. Therefore, for each subsequence, we can conclude that $F_{m_h}$ approaches F. Therefore, $F_m$ also converges to F. This means that

$$\begin{array}{c}\hfill \underset{\alpha \to 0}{lim}\frac{{\lambda }_1\left(\alpha \right)-{\lambda }_1\left(0\right)}{\alpha }=F,\end{array}$$

exists. Consequently, ${\lambda }_1$ is differentiable at $\alpha =0$ and ${\dot{\lambda }}_1\left(0\right)=F$. This proves the statement. □