Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard

We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the Hausdorff dimension of the non-wandering set of the billiard. Finally we show that the Hausdorff dimension depends differentiably on sufficiently smooth deformations to the boundary of the billiard, and estimate the derivative with respect to such deformations.


Introduction
The dimension theory of dynamical systems studies the dimensional characteristics (such as Hausdorff dimension) of the invariant sets of dynamical systems. See [P] for an introduction to the theory, or [BG] for a recent review of this field. Past work has examined how dimensional characteristics of various dynamical systems can change with respect to perturbations of the system, for example the differentiability of entropy of Anosov flows [KKPW], SRB measures in hyperbolic flows [R], and Hausdorff dimension of horseshoes [M]. However this kind of problem has not been considered in the context of open billiard systems. In this paper we show that the Hausdorff dimension of the non-wandering set for an open billiard in the plane depends differentiably on perturbations to the boundary of the billiard.
A billiard is a dynamical system in which a single pointlike particle moves at constant speed in some domain Q ⊂ R n and reflects off the boundary ∂Q according to the classical laws of optics [S]. Open billiards are a class of billiard in which the domain Q is unbounded. Let K = K 1 ∪ . . . ∪ K m (m ≥ 3) be a subset of R 2 , where K i are compact strictly convex disjoint domains in R 2 with C r boundaries (r ≥ 3). Set Q = R 2 \K. We assume that K satisfies the "no-eclipse condition" introduced by Ikawa in [I]: (H) For 1 ≤ i, j, k ≤ m with i = j = k, the convex hull of K i ∪ K j is disjoint from K k .
This condition ensures that the collision angle φ is bounded above by a constant φ max < π 2 , and prevents discontinuities in the non-wandering set M 0 which consists of all bounded billiard trajectories in Q.
In this paper we consider smooth homotopies between different billiards with a parameter α which we call billiard deformations. Section 2 contains some preliminaries on open billiards and a precise definition of the deformations. In section 3 we show that the periodic trajectories in the non-wandering set are differentiable and Lipschitz with respect to α. In section 4 we extend this to the whole non-wandering set, and show that the curvature of the stable and unstable manifolds are also differentiable and Lipschitz.
The Hausdorff dimension of the non-wandering set was estimated in [K] by investigating convex fronts. This was later improved and extended to higher dimensional billiards in [Wr]. In section 5, using techniques from Pesin's book on dimension theory in dynamical systems [P], we recover the estimates in [K] and show that they also apply to the lower and upper box dimensions. That is, where d min , d max , k min , k max are constants that depend on simple geometric properties of the obstacles. Furthermore we show that these dimensions depend differentiably on the boundary of the billiard obstacles. That is, if the billiard is shifted or deformed smoothly with some parameter α, then the function where C is a constant depending only on simple geometrical properties of the obstacles. Similar results have been achieved for other dynamical systems (see e.g. horseshoes in [M]), but this is the first time it has been done for nonwandering sets of open billiards.

Open billiards
Consider the set Q described in the introduction. We describe a particle in the billiard by x t = (q t , v t ) where q t ∈ Q is the position of the particle and v t ∈ S 1 is its velocity at time t. Then for as long as the particle stays inside Q, it satisfies Collisions with the boundary are described by where n is the normal vector (into Q) of ∂Q at the point of collision, v − is the velocity before reflection and v + is the velocity after reflection.

Non-wandering set
For x = (q, v) with q ∈ ∂K, v ∈ S 1 , we denote by n = n K (q) the outward unit normal vector of ∂K at q, by φ(x) the angle between v and n (bounded by φ max < π 2 ), by κ(q) the curvature of ∂K at q, and by t j (q, v) ∈ [−∞, ∞] the time of the j-th reflection of x. If the forward trajectory of x does not have at least j reflections, then t j (x) = ∞, and if the backward trajectory does not have at least j reflections then t j ( . Then B is invertible and C r−1 . On the tangent space T x M , for x ∈ M , we will use the norm (dq, dv) = cos φ|dq| (see e.g. [CM]).
The set M together with an inner product inducing this norm is a Riemannian manifold. The non-wandering set of a billiard is the set of points whose trajectories are bounded. The non-wandering set of the billiard flow is denoted Ω (S) or Ω, and its restriction to the boundary of K is M 0 = Ω ∩ (∂K × S 1 ). Equivalently, M 0 = {x ∈ M : |t j (x)| < ∞, ∀j ∈ Z} is the non-wandering set of the billiard map B.

Billiard deformations
Here we define precisely what we mean by deformations to the boundary. We will always assume the boundaries ∂K i of each obstacle are parametrised counterclockwise. We will say that a function of two variables f (x, y) is called C A,B -smooth if for every a ≤ A, b ≤ B the derivatives ∂ a+b f ∂x a ∂y b are continuous. Definition 2.1. Let I be a compact interval in R. For any α ∈ I, let K = K(α) be a subset of R 2 . We call K(α) a billiard deformation if there exist integers r ≥ 3, r ′ ≥ 1, bounded functions l i (α) > 0, C r,r ′ -smooth functions ϕ i : I × [0, l i ] → R 2 , and positive constants s max , s min , a max , a min , c 1 , c 2 , c 3 , c 4 , such that for all α ∈ I and u i ∈ [0, l i (α)], the following conditions hold: is an open billiard satisfying (H).
i ∂α ≤ c 4 . We call α the deformation parameter. If the obstacles are parametrized by arc length then we have s min = s max = 1, a min = κ min , a max = κ max , and ∂ 2 ϕi ∂u 2 i = κ(ϕ i ). Since it is always possible to reparametrize the boundary by arc length, and this makes the proofs significantly easier, from now on we will assume that each boundary is parametrized by arc length.
If K(α) simply shifts one or more of the obstacles, e.g. K i (α) = K i + αv for a constant vector v, then we have c 2 = c 4 = 0. Sometimes a deformation will affect only some of the obstacles. Define a function δ i such that

Symbolic model
. This function is C r with respect to each u j and differentiable with respect to α.
These points determine a billiard trajectory that satisfies the classical law of optics.
This shows that the representation map f is invertible and its inverse is where v 12 is the unit vector from p 1 to p 2 , and the p i are found by minimizing the length function.
For any x, let κ j = κ(πB j x) be the curvature at πB j x, φ j = φ(B j x) be the angle between the velocity vector and the normal vector of B j x, and let γ j = 2κj cos φj . Let d min , κ min and γ min be the minimum values of d 1 (x), κ 0 (x) and γ 0 (x) respectively over all x ∈ M 0 , and let d max , κ max and γ max be the respective maximum values. Note that γ min = 2κ min . Also recall that φ max < π 2 is the maximum value of φ(x) over x ∈ M 0 . Whenever we are considering a fixed sequence ξ we will use the abbreviation ϕ j = ϕ ξj .

Derivatives of periodic points
In this section we prove the following theorem.
Theorem 3.1. Let K be an open billiard satisfying the no eclipse condition, and let T α be a billiard deformation. For any sequence ξ with period n, if the corresponding periodic points are given by p j = ϕ ξj (u j (α), α), then the parameters u j (α) are differentiable with respect to the deformation parameter α and the derivatives satisfy Proof. Fix a sequence ξ with period n. We will use the following notation:

Total derivative with respect to α
For each j, we have the equation This has a single critical point u(α) = (u 1 (α), . . . , u n (α)), which satisfies We can take the total derivative with respect to α to get a system of equations, Denoting the Hessian matrix of G by H, we have the matrix equation The Hessian matrix is non-singular, (see [St]), so by the implicit function theorem, the solutions ∂uj ∂α to this system exist. Since ϕ is C r,r ′ , we can conclude that H is C r−2,r ′ , ∂ ∂α ∇G is C r−1,r ′ −1 , and hence ∂u ∂α is C r−2,r ′ −1 . We will use the following vector identity several times. If u, v, w are unit vectors in the plane, then where v ⊥ is a positive (counterclockwise) rotation by a right angle.

Estimating − ∂ ∂α ∇G
For any i = 1, . . . n, let δ i = 1 if K ξi is affected by the billiard deformation and 0 otherwise. Note that v jj−1 + v jj+1 = −2 cos φ j n j where n j is the normal vector of K ξj at φ j . We also use the vector identity (2).

The Hessian Matrix
The Hessian of G is a matrix composed of the derivatives ∂ 2 G ∂uj ∂ui . This section follows [St], see also [PS]. The first derivatives can be written If i ∈ I j , we can use (2) to get Along the diagonal i = j we have Note that v jj−1 + v jj+1 = −(2 cos φ j )n j , where n j is the outward unit normal vector. So we have i∈Ij v ji , ∂ 2 ϕ j ∂u 2 j = 2κ j cos φ j . Using the vector identity (2) we get Finally, if i / ∈ I j ∪ {j}, then ∂ 2 G ∂uj ∂ui = 0. We will now show the derivatives ∂uj ∂α are bounded.

Solving the cyclic tridiagonal system
We now have the following system of equations: Make the substitutions y j = ∂uj ∂α cos φ j , γ j = 2κj cos φj and a j = a jj−1 , a 1 = a n1 . Then we can rearrange the system to: We can write this as a matrix equation Ay = b, where y = (y 1 , . . . , y n ) ⊺ , b = (b 1 , . . . , b n ) ⊺ , and A is the following matrix: . . a n a 1 0 a n a n + a 1 + γ n A tridiagonal matrix only has non-zero elements in the main diagonal and the first diagonals above and below the main diagonal. A is cyclic tridiagonal, meaning it has two more nonzero elements in the corners. This matrix is also diagonally dominant by rows since a j + a j+1 + γ j > a j + a j+1 > 0. We can estimate where λ min is the smallest eigenvalue of A. However since √ n is unbounded we cannot use this to find constant bounds on y j that hold for all n. Instead we use the following theorem of Varah [Va]. Let ∞ denote the matrix norm induced by the infinity norm.

Theorem 3.2. [Va] Let
For our matrix, we have

Returning to the system Ay
Recalling the substitutions we made earlier, we get Corollary 3.3. Recall that the periodic points p ξ = (p 1 , . . . , p n ) are given by p j = ϕ ξj (u j (α), α). So each p j is differentiable with respect to α and we have where δ i = 1 if K i is affected by the deformation and 0 otherwise.
Proposition 4.2. Let I be an interval and X be a set. Then for any sequence of functions g n : I → X, if g n converges pointwise to g and g ′ n converges uniformly to h, then g is differentiable and g ′ = h.
Let ξ ∈ Σ and define a sequence of periodic sequences {ξ (n) } n in Σ by ξ (n) j = ξ (j mod n) , so that ξ and ξ (n) are on the same n-cylinder. From Proposition 4.1, the points p α (ξ (n) ) converge uniformly (with respect to α) to p α (ξ) as n → ∞. For ξ(n), we have The elements of this equation can all be expressed (using inverse functions and equation (1)) as continuous functions of the point p α (ξ (n) ). Since p α (ξ (n) ) converges uniformly, by basic limit properties dpα(ξ (n) ) dα must converge uniformly for all α ∈ I. Treating p α (ξ (n) ) as a sequence of functions of α, Proposition 4.2, implies that p α (ξ) is differentiable and Now by combining this with Corollary 3.3, for any ξ ∈ Σ we have So every point in the non-wandering set is differentiable with respect to deformations of the billiard, and the derivative is bounded.

Estimating derivatives of distances, curvatures and collision angles
We can use the upper bound on ∂uj ∂α to estimate the derivatives of other properties of billiard trajectories, specifically the distances d j , curvature κ j and angles φ j . Let C 1 = 1 cos φj c1+c2dmin κmindmin . Then The derivative of κ j can be bounded using the following billiard constant.

Recall that
j ∂α are bounded above by c 3 and c 4 . Then κ j is differentiable with respect to α and The collision angle φ j satisfies cos 2φ j = We also use the expression γ j = 2κj cos φj in this paper. This is also differentiable and we have

Stable and unstable manifolds
The billiard map B and the flow S t are examples of an axiom A diffeomophism and an axiom A flow respectively. It is well known that for any point x ∈ M 0 there exist local stable and unstable manifolds W The stable manifold is simply the time reversal of the unstable manifold, that is W where Refl:Q →Q is a bi-Lipschitz involution given by for q ∈ ∂K.
The billiard map in the plane is trivially a conformal map in the following sense.
Definition 4.3. [P] An axiom A diffeomorphism f with a basic set Λ is called u-conformal (respectively, s-conformal) if there exists a continuous function where Isom x is an isometry of E (u) or E (s) . Then f is called conformal if it is both u-conformal and s-conformal.
For higher dimensional billiards, B is not conformal in general. We now define the convex fronts used to calculate and differentiate the functions a (u) and a (s) .
Definition 4.4. Let x = (q, v) ∈ Ω and let x 0 be the unique point on ∂K such that S t x 0 = x for some t ≥ 0. Let X(x) ⊂ Ω be the unique convex curve containing q such that for any y ∈ W (u) ε (x 0 ), there exists t ≥ 0 and p ∈ X such that S t (y) = (p, n X (p)). Then define k(x) to be the curvature of X(x) at x. X(x) is a C r curve and the map x → X(x) is at least C 1 in general [S, CM]. However if y ∈ X(x) then the curve X(y) overlaps with X(x), so x → X(x) is C r when restricted to these curves. The restriction of k to a curve X(x) is C r−2 since curvature involves the second derivative.
Proposition 4.5. [S] (see also [CM]) Let K be a planar open billiard, and ε (x). Then the billiard map B is conformal on its stable and unstable manifolds, with functions a (u) Note that |a (s) (x)| < 1 and |a (u) (x)| > 1 for all x ∈ M 0 .

Curvature of unstable manifolds
Let x ∈ M 0 and let k j (x) = k(B j x) be the curvature of the convex front X(B j x) at B j x. We have the following well-known reccurance relation for k j [S] (see also [CM]).
We can use this to obtain bounds 2κ min ≤ k j ≤ 1 dmin + 2κmin cos φmax . In [K, Wr] these bounds can be improved to 2κ min < k min ≤ k j ≤ k max < 1 dmin + 2κmin cos φmax , where k min , k max are constants derived from the billiard constants. If x is periodic with period n, then it is possible to solve for k(x) using these equations together with k n = k 0 . Furthermore, by writing x = p α ξ we can differentiate with respect to α to get For any periodic sequence ξ, we can write x = p α ξ and differentiate the function log a (u) (x) with respect to α. We have Since the periodic orbits are dense in M 0 , and x → a (u) (x) is C 1 , we have 5 Derivative of Hausdorff dimension

Entropy and pressure
Let X be a compact metric space, f : X → X a continuous map, Λ ⊂ X a hyperbolic f -invariant subset, and ψ : X → R a continuous function. Denote by M(X) the set of all f -invariant Borel ergodic measures on X. Let h µ (f ) denote the topological entropy with respect to a measure µ ∈ M(X), and let P (ψ) = P Λ (ψ) denote the topological pressure, as defined as in [P] or [Wa]. The variational principle is There is a unique equilibrium measure µ = µ(ψ) corresponding to ψ that satisfies Pressure and entropy can also be defined using operators on the symbol space Σ + (see [PP]). For a Lipschitz function ψ ∈ C(Σ + ), that is ψ : Σ + → R, define the Ruelle operator L ψ : C(Σ + ) → C(Σ + ) by (L ψ w)(x) = σy=x e ψy w(y).
Then L ψ is a bounded linear operator. The Ruelle-Perron-Frobenius theorem guarantees a simple maximum positive eigenvalue β for L ψ . We define P (ψ•p) = log β, and this definition is equivalent to the definition above. The entropy h µ (f ) is then defined as P (0). Proof. This can be easily calculated using the Ruelle-Perron-Frobenius theorem.

Bowen's equation
Let Λ be a locally maximal hyperbolic set for a conformal, topologically mixing, C 2 diffeomorphism f on a Riemannian manifold M . Following [P], let t (u) , t (s) be the unique roots of Bowen's equation, where a (u) and a (s) are defined as in Definition 4.3 Denote by κ (u) and κ (s) the unique equilibrium measure corresponding to the functions −t (u) log |a (u) (x)| and t (s) log |a (s) (x)| respectively. Then from the variational principle we have We will use the following theorem by Pesin.
Theorem 5.2. [P] Let f be a C 2 diffeomorphism with a locally maximal hyperbolic set Λ. If f is u-conformal then for any x ∈ Λ and any open set Similarly, if f is s-conformal then for any x ∈ Λ and any open set S ⊂ W (s) (x) such that S ∩ Λ = ∅, Finally, if f is u-conformal and s-conformal, the dimensions of Λ are simply dim H (Λ) = dim B (Λ) = dim B (Λ) = t (u) + t (s) .
(4) So using Theorem 5.2, we obtain the following theorem.
Theorem 5.3. Let K be an open billiard with C 3 boundary satisfying the condition (H). Then the Hausdorff and upper and lower box dimensions of the nonwandering set are equal and given by 2 log(m − 1) log(1 + d min k min ) , 2 log(m − 1) log(1 + d max k max ) .
This is consistent with the results in [K] and [Wr].
estimating the periodic points would require solving a cyclic block tridiagonal matrix. This is not too difficult and will likely give similar estimates for the points. However since the billiard map B is not conformal in higher dimensional billiards, there is currently no known equation for the Hausdorff dimension of the non-wandering set (see [BG] for some of the latest results in this area). Instead we can find upper and lower bounds related to the functions ψ (u) (x) = − log d x B|E (u) and ψ (u) (x) = log (d x B) −1 |E (u) [B]. Without an exact equation to differentiate, it will be hard to determine whether the Hausdorff dimension for non-planar billiards is differentiable or not.