Hyperbolic billiards on polytopes with contracting reflection laws

We study billiards on polytopes in $\Rr^d$ with contracting reflection laws, i.e. non-standard reflection laws that contract the reflection angle towards the normal. We prove that billiards on generic polytopes are uniformly hyperbolic provided there exists a positive integer $k$ such that for any $k$ consecutive collisions, the corresponding normals of the faces of the polytope where the collisions took place generate $\Rr^d$. As an application of our main result we prove that billiards on generic polytopes are uniformly hyperbolic if either the contracting reflection law is sufficiently close to the specular or the polytope is obtuse. Finally, we study in detail the billiard on a family of $3$-dimensional simplexes.


1.
Introduction. Given a d-dimensional polytope P , a billiard trajectory inside P is a polygonal path described by a point particle moving with uniform motion in the interior of P . When the particle hits the interior of the faces of P , it bounces back according to a reflection law. Therefore, a billiard trajectory is determined by a sequence of reflections on the faces of P . Any reflection can be represented by a pair x = (p, v) where p is a point belonging to a face of P and v is a unit velocity vector pointing inside P . We denote by M the set of reflections. The map Φ : M → M, x → x that takes a reflection x to the next reflection x is called the billiard map. The dynamics of billiards on polytopes has been mostly studied considering the specular reflection law. More recently, in the case of polygonal billiards, a new class of reflection laws has been introduced that contract the reflection angle towards the normal of the faces of the polygon [1,8,2,4]. These are called contracting reflection laws. A billiard map with a contracting reflection law is called a contracting billiard map. It is known that strongly contracting billiard maps on generic convex polygons are uniformly hyperbolic and have finite number of ergodic SRB measures [5]. Recently, it has been proved that the same conclusion hods for contracting billiard maps on polygons with no parallel sides facing each other (even for contracting reflection laws close to the specular and for non-convex polygons) [7].
In this paper we extend some of the previous results to contracting billiard maps on polytopes. It is known that the contracting billiard map of any polygon has dominated splitting [5,Proposition 3.1]. In this direction we show in Proposition 4 that the contracting billiard map of any polytope is always (uniformly) partially hyperbolic, i.e. there is a continuous and invariant splitting E s ⊕ E cu of the tangent bundle of M into subbundles of the same dimension such that DΦ uniformly contracts vectors in the stable subbundle E s and has neutral or expanding action on vectors belonging to the centre-unstable subbundle E cu .
There are essentially two obstructions for the uniform expansion in the centreunstable subbundle E cu . The first obstruction is caused by the billiard orbits that get trapped in a subset of faces of P whose normals do not span the ambient space R d . When P is a polygon (d = 2), those orbits are exactly the periodic orbits of period two, i.e. orbits bouncing between parallel sides of P . In fact, when P has no parallel sides the contracting billiard map is uniformly hyperbolic [5,Proposition 3.3]. As another example let P be a 3-dimensional prism and consider a billiard orbit unfolding in some plane parallel to the prism's base. The normals to the faces along this orbit will span a plane and the billiard map behaviour transversal to this plane is neutral. This leads to an expansion failure in E cu .
In order to circumvent this obstruction we had to consider a class of polytopes which have the property that for any subset of d faces of P the corresponding normals span R d . A polytope with this property is called spanning (see Definition 2.9). In addition to being spanning, we suppose that the normals to the (d − 1)-faces incident with any given vertex are linearly independent (see Definition 3.1). Spanning polytopes with these properties are generic. In fact they form an open and dense subset having full Lebesgue measure in the set of all polytopes.
The second obstruction to uniform expansion corresponds to the billiard orbits that spend a significant amount of time bouncing near the skeleton of P . To control the time spent near the skeleton we introduced the notion of escaping time. Roughly speaking, the escaping time of x ∈ M is the least positive integer T = T (x) ∈ N ∪ {∞} such that the number of iterates it takes for the billiard orbit of x to leave a neighbourhood of the skeleton of P is less than T (see Definition 2.11).
With these notions we prove that the contracting billiard map has non-zero Lyapunov exponents for almost every point with respect to any given ergodic invariant measure. More precisely we prove: Theorem 1.1. If the contracting billiard map Φ of a generic polytope has an ergodic invariant probability measure µ such that T is integrable with respect to µ, then µ is hyperbolic.
When the contracting billiard map Φ has bounded escaping time, then Φ is uniformly hyperbolic. Theorem 1.2. If the contracting billiard map Φ of a generic polytope has an invariant set Λ such that T is bounded on Λ, then Φ| Λ is uniformly hyperbolic. Theorems 1.1 and 1.2 follow from Theorem 2.10 which gives a uniform estimate on the expansion along the orbit of every point which is k-generating (see Definition 2.8). Being k-generating simply means that the face normals along any orbit segment of length k span R d .
The strategy to prove Theorem 2.10 is the following. Consider the billiard orbit x n = (p n , v n ), n ≥ 0 of a k-generating point x 0 = (p 0 , v 0 ) ∈ M . Denote by η n the inward unit normal of the face of P where the reflection x n takes place. In some appropriate coordinates, known as Jacobi coordinates, the unstable space E u (x 0 ) is represented by the orthogonal hyperplane v ⊥ 0 . If the velocity v 1 is collinear with the normal η 1 , then the action of the derivative DΦ on E u (x 0 ) is neutral. Otherwise, the map DΦ expands the direction v ⊥ 0 ∩ V 1 where V 1 denotes the plane spanned by the velocities v 0 and v 1 . Similarly, DΦ 2 expands the directions contained in v ⊥ 0 ∩ V 2 where now V 2 is generated by the velocities v 0 , v 1 and v 2 . However, it may happen that the plane spanned by the velocities v 1 and v 2 is the same obtained from the span by the normals η 1 and η 2 , thus implying that dim(v ⊥ 0 ∩ V 2 ) = 1. This coincidence of the velocity front with the normal front is called a collinearity (see Definition 5.6).
If a collinearity never occurs and x 0 is k-generating then the map DΦ k expands d − 1 distinct directions in v ⊥ 0 . Although collinearities prevent full expansion of the iterates DΦ n (x 0 ) they have the good trait of synchronizing the velocity front with the normal front. After a collinearity every time a new face is visited the angle between the new velocity and the previous velocity front is always bounded away from zero. This happens because this velocity angle is related to the angle between the new normal and the previous normal front, and also because we assume the polytope to be spanning. Consider now the velocity front V at some collinearity moment. The previous property implies expansion of DΦ n (x 0 ) transversal to v ⊥ 0 ∩V after the collinearity moment. Choosing a minimal collinearity (see Definition 5.7) in the orbit of x 0 we can also ensure the expansion of DΦ n (x 0 ) along the velocity front up the collinearity moment. Putting these facts together, if at some instant t < k a minimal collinearity occurs on the orbit of x 0 then for n ≥ t + k we have full expansion of DΦ n (x 0 ) on E u .
Because we seek uniform expansion, one has to deal with δ-collinearities instead (see Definition 5.9). Moreover, since the set of orbits in M is not compact (one has to remove from M the orbits which hit the skeleton of P ), δ-collinearities are more easily handled in a bigger set called the trajectory space. The trajectory space is compact and defined in a symbolic space which only retains the velocities and the normals of the faces of P where the reflections take place (see Definition 5.1). Finally, using compactness and continuity arguments we derive Theorem 5.3 which gives a uniform estimate on the expansion along an orbit segment of length 2k of any k-generating point. Then Theorem 2.10 follows immediately from Theorem 5.3. The crucial tool to prove Theorem 5.3 is Lemma 5.14 which gives a uniform expansion estimate on compositions of linear maps. Since this lemma is formulated in more conceptual terms, we believe that the ideas therein might be of independent interest.
In section 4 we show that contracting billiards on polytopes have finite escaping time if either the contracting law is close to the specular or the polytope is obtuse. This together with Theorem 1.2 prove the following corollaries. Corollary 1. The contractive billiard map of a generic polytope with a contracting reflection law sufficiently close to the specular one is uniformly hyperbolic. Corollary 2. The contracting billiard map of a generic obtuse polytope is uniformly hyperbolic.
The rest of the paper is organized as follows. In section 2 we introduce some notation and define the contracting billiard on a polytope. We also derive several properties of contracting billiards maps and rigorously state our main result. In section 3 we show that polytopes on general position are generic and in section 4 we study the escaping time on polyhedral cones. Technical results concerning the expansion of composition of linear maps are proved in section 5. In section 6 we prove our main results. Finally, in section 7 we study in detail the contracting billiard of a family of 3-dimensional simplexes.

Definitions and statements.
for some non-zero vector v ∈ R d and some real number c ∈ R. A polyhedron is any finite intersection of half-spaces in R d . A polytope is a compact polyhedron. We call dimension of a polyhedron to the dimension of the affine subspace that it spans. Let P ⊂ R d be a d-dimensional polytope.
The billiard on P is a dynamical system describing the linear motion of a point particle inside P . When the particle hits the boundary of P , it gets reflected according to a reflection law, usually the specular reflection law. In the following we rigorously define the billiard map Φ P with the specular reflection law. But first, let us introduce some notation.
2.1. Basic Euclidean geometry. Let V and V be Euclidean spaces with dim V = dim V = d. Given a linear map L : V → V , the maximum expansion of L is the operator norm while the minimum expansion of L, defined by is either 0, when L is non invertible, or else m(L) = L −1 −1 . We denote by L * : V → V the adjoint operator of L : V → V . Recall that the singular values of L are the eigenvalues of the conjugate positive semi-definite symmetric operator √ L * L. Being real, and non negative, the singular values of L can be ordered as follows This determinant is the factor by which L expands d-volumes.
Given λ > 0 we denote by v ≥ λ (L) the direct sum of all singular directions of L (eigen-directions of L * L) associated with singular values µ ≥ λ. Likewise, we denote by v < λ (L) the direct sum of all singular directions of L associated with singular values µ < λ. It follows from these definitons that and similar relations hold for L * . To shorten notations we will simply write v(L) instead of v ≥ L (L). This subspace will be referred to as the most expanding direction of L.
Given vectors v 1 , . . . , v n ∈ R d , the linear subspace spanned by the vectors v 1 , . . . , v n is denoted by v 1 , . . . , v n . Let S denote the unit sphere in Let v, η ∈ S be unit vectors and u ∈ R d . We denote by S + η the hemisphere associated with η, Let η ⊥ denote the orthogonal hyperplane to η. The orthogonal projection of u onto the hyperplane η ⊥ is, where P η (u) = u, η η, is the orthogonal projection of u onto the line spanned by η.
The reflection of u about the hyperplane η ⊥ is defined by, Finally, the parallel projection of u along v onto the hyperplane η ⊥ is Denote by ∠(v, w) the angle between two non-zero vectors in R d , defined as The angle between a non-zero vector v ∈ R d and a linear subspace The angle between two linear subspaces E and F of R d of the same dimension is defined as This angle defines a metric on the Grassmann manifold Gr k (R d ) of all k-dimensional linear subspaces E ⊆ R d . Given two linear subspaces E, F ⊆ R d , with dim E ≤ dim(F ⊥ ), we define the minimum angle Unlike the previous angle, this minimum angle is not even a pseudo-metric on The minimum angle ∠ min (E, F ) quantifies the 'transversality' on the intersection E ∩ F .
We denote by π E,F ⊥ : E → F ⊥ the restriction to E of the orthogonal projection to F ⊥ .
Since dim E = dim F , there is an orthogonal linear map Φ : which implies that π E,F ⊥ = π F,E ⊥ . Thus the sine of the maxima in the definition of ∠(E, F ) coincides with this common norm.
Lemma 2.2. Let E, E and H be linear subspaces of R d such that Then Proof. First notice that .
On the last equality we use that v = h + u is an orthogonal decomposition with h ∈ H and u ∈ H ⊥ . Thus taking the sup in u ∈ (E + H) ∩ H ⊥ \ {0} we get Equality holds when dim E = 1.
Proof. Just notice that all singular values of π E,F ⊥ are in the range [0, 1] because π E,F ⊥ is the restriction of an orthogonal projection.
Given an integer k ∈ N and a linear subspace E ⊆ R d , the Grassmann space of k-vectors in E will be denoted by ∧ k (E). This space inherits a natural Euclidean structure from E (see [10]). . . , f r } respectively such that k ≤ d − r. Let e = e 1 ∧ . . . ∧ e k ∈ ∧ k (E) and f = f 1 ∧ . . . ∧ f r ∈ ∧ r (F ). Then Proof. Given a unit vector v ∈ E, by the proof of Lemma 2.1 we have sin On the other hand because e = f = 1. The middle inequality follows from Lemma 2.3.
Proof. Let {e 1 , . . . , e r } be an orthonormal basis of E. We apply Lemma 2.4 to the subspaces v i and E ⊕ v 1 , . . . , v i−1 . Since the first subspace has dimension 1 the inequality in this lemma becomes an equality. Hence, because v i = 1 we have Multiplying these inequalities and using Lemma 2.4 again we obtain We have used above that v 1 ∧ . . . ∧ v k ≤ v 1 · · · v k = 1.

2.2.
Billiard map. Suppose that P has N faces (of dimension d − 1) which we denote by F 1 , . . . , F N . For each i = 1, . . . , N , denote by η i the interior unit normal vector to the face F i . Also denote by Π i the hyperplane that supports the face F i . We write the interior of The domain of the billiard map Φ P is the set of points (p, v) ∈ M such that the half-line { p + t v : t ≥ 0} does not intersect the skeleton ΣP . We denote this set by M . Clearly, M is the complement of a co-dimension two subset of M . Now the billiard map Φ P : M → M is defined as follows. Given x = (p, v) ∈ M , let τ = τ (p, v) > 0 be minimum t > 0 such that p + t v ∈ F j for some j = 1, . . . , N .

PEDRO DUARTE, JOSÉ PEDRO GAIVÃO AND MOHAMMAD SOUFI
The real number τ is called the flight time of (p, v). Then the billiard map is defined by ). Note that the billiard map Φ P is a piecewise smooth map and it has finitely many domains of continuity. The number of domains of continuity is at most N (N − 1), which is the number of 2-permutations of N faces. If P is convex, then all permutations define a branch map.
It is easy to obtain a formula for the branch maps and its derivatives.
Taking the inner product with η j in both sides of the equation and noting that p − p j , η j = 0, we get To prove the formula for the derivative, define the map Ψ η : (p, v) → P v,η ⊥ (p) for any given η ∈ S. The claim follows from the formula

Contracting reflection laws. A contracting law is any family
A contracting law can be uniquely characterized by a single C 2 map of the interval 0, π 2 as the following proposition shows. Proposition 2. Given a contracting law { C η : where θ = arccos v, η is the angle between η and v, (d) for every η ∈ S, Proof. Let η ∈ S and v ∈ S + η . By item (b) of the definition of a contracting law we can write where a η and b η are non-negative C 2 functions. Taking the inner product with η on both sides of the previous equation we get, is the angle formed by the vectors v and η. By This shows (c). The remaining properties follow immediately.

Contracting billiard map. Given a contracting law {C
where η(p) denotes the interior unit normal of the face of the polytope where p lies. The There is a system of coordinates which is convenient to represent the derivative of the contracting billiard map.
The previous linear isomorphism will be referred as Jacobi coordinates on the tangent space T x M . We shall use the notation (J, J ) to denote an element in v ⊥ × v ⊥ .
The following proposition gives a formula for the derivative of the contracting billiard map in terms of Jacobi co-ordinates.
2.5. Orbits, invariant sets and hyperbolicity. Denote by M + the subset of points in M that can be iterated forward, i.e.
A billiard path or trajectory is the polygonal path formed by segments of consecutive points of a billiard orbit. Define It is easy to see that D is an invariant set and Φ f,P and its inverse are defined on D. Following Pesin we call the closure of D the attractor of Φ f,P . We say that Λ ⊂ M is an invariant set if Λ ⊂ D and Φ −1 f,P (Λ) = Λ. To simplify the notation let us write Φ = Φ f,P .
Definition 2.6. Given an invariant set Λ of Φ, we say that Φ is uniformly partially hyperbolic on Λ if for every x ∈ Λ there exists a continuous splitting and there are constants λ < 1, σ ≥ 1 and C > 0 such that for every n ≥ 1 we have If σ > 1, then we say that Φ is uniformly hyperbolic on Λ and write E u for the subbundle E cu . When Λ = D, then we simply say that Φ is uniformly partially hyperbolic.
We denote by The proof of the following result is an adaptation of [5, Proposition 3.1].
Proposition 4. For any polytope P and any contracting reflexion law f , Φ f,P is uniformly partially hyperbolic.
A simple computation shows that We claim that writing x n = (p n , v n ) = Φ n x and denoting by Z n the zero endomorphism on v ⊥ n , the following limit exists A recursive computation allows to explicit the right hand side composition in the previous limit, which is a partial sum of the following series the previous facts show that Φ is uniformly partially hyperbolic.
2.6. Main results. Definition 2.8. Given k ∈ N, we say that x ∈ M + is k-generating if the face normals along any orbit segment of length k of the orbit of x generate the Euclidean space R d . Definition 2.9. Given ε > 0, the polytope P is called ε-spanning if for any d distinct faces F i1 , . . . , F i d of P with interior normals η i1 , . . . , η i d , the angle between η i1 and E := η i2 , . . . , η i d is at least ε, i.e.
We also say that P is a spanning polytope if it is ε-spanning for some ε > 0.
The following theorem is the main result of this paper. It shows that the contracting billiard map uniformly expands the unstable direction along the orbit of any k-generating point. Moreover, the expanding rate only depends on the polytope and contracting reflection law.
Theorem 2.10. Suppose P is a spanning polytope and f a contracting reflexion law. There exists σ = σ(f, P ) > 1, depending only on f and P , such that for every k ≥ d and every k-generating x ∈ D, We prove this theorem and the following results in section 6.
Definition 2.11. Given x ∈ M + , the escaping time of x, denoted by T (x), is the least positive integer k ∈ N such that x is k-generating. If x is not k-generating for any k ∈ N, then we set T (x) = ∞. We also call the function T : M + → N ∪ {∞} the escaping time of P with respect to f . Theorem 2.12. Suppose P is a spanning polytope and µ is an ergodic Φ f,Pinvariant Borel probability measure. If T is µ-integrable, then µ is hyperbolic.
The concept of polytope in general position, mentioned in the following corollaries, is defined below (see definition 3.1).
Corollary 3. Suppose P is a polytope in general position. There exists λ 0 = λ 0 (P ) > 0 such that for every contracting reflection law f satisfying λ(f ) > λ 0 the billiard map Φ f,P is uniformly hyperbolic.
A polytope P in general position is called obtuse if the barycentric angle at every vertex of P is greater than π/4 (see section 4 for a precise definition).
Corollary 4. Suppose P is a polytope in general position and f any contracting reflection law. If P is obtuse, the Φ f,P is uniformly hyperbolic.
Consider the class P N of d-dimensional polyhedra P ⊂ R d that contain the origin, i.e., 0 ∈ int(P ), with exactly N faces. Given N points (p 1 , . . . , p N  is open in (R d \ {0}) N , and the range of Q : U → · coincides with P N . Locally the map Q : U → P N is one-to-one, and determines an atlas for a smooth structure on P N . We will consider on this manifold the Lebesgue measure obtained as pushforward of the Lebesgue measure on (R d \ {0}) N by the map Q.
Let P N denote the subset of polytopes in P N . In Algebraic Geometry, the following result is a standard consequence of the notion of 'general position'. We include its proof here for the reader's convenience, also because we could not find any reference for this precise statement. Proof. Consider the subsets N 1 ⊂ P N , resp. N 2 ⊂ P N , of polytopes where condition (1), resp. (2), of definition 3.1 is violated. It is enough to observe that the sets N 1 and N 2 are finite unions of algebraic varieties of co-dimension one. For Then N 2 is covered by the union over all 1 ≤ i 1 < i 2 < . . . < i d+1 ≤ N of the hypersurfaces defined by the algebraic equation In fact, if there is a point x 0 ∈ R d in the intersection of d + 1 distinct hyperplanes then the matrix with rowsp i1 ,p i2 , . . . ,p i d+1 contains the vector (x 0 , −1) ∈ R d+1 in its kernel, which implies (1). Analogously, N 1 is contained in the union over all 1 ≤ i 1 < i 2 < . . . < i d ≤ N of the hypersurfaces defined by the algebraic equation 4. Escaping times. In this section we study the escaping times of billiards on polyhedral cones with contracting reflection laws.
Let Π 1 , . . . , Π s be s hyperplanes in R d passing through the origin. For each hyperplane Π i we take a unit normal vector η i and we suppose that the set of hyperplanes are in general position, i.e. the normal vectors η 1 , . . . , η s are linearly independent. A set of s hyperplanes in general position define a convex polyhedral cone . . , s} . For polyhedral billiard with the specular reflection law, Sinai proved that there exists a constant K > 0, depending only on Q, such that every billiard trajectory in Q has at most K reflections [9]. In this case we say that Q has finite escaping time.
By projecting the billiard dynamics to the orthogonal complement of s i=1 Π i , we may assume that the normal vectors η 1 , . . . , η s defining the polyhedral cone Q span R d . Thus, from now on we set s = d. Associated with a convex polyhedral cone Q there is a constant measuring the aperture of Q. It is defined as follows. where is the distance of H to the origin. The barycentric angle φ of Q is defined by sin φ = (see Figure 1). Note that 0 < φ < π/2. We say that a convex polyhedral cone Q is obtuse if φ > π/4.
where u k = R ηi k (v k ), θ k = arccos u k , η i k and η i k is the inward normal of P where the k + 1-th collision took place.
for every k ≥ 0.
Given a sequence of consecutive reflection velocities v 0 , . . . , v n we denote by L the length of the zigzag path formed by the reflections, i.e.
We say that Q has bounded zigzag reflections if there exists a constant C > 0 such that L(v 0 , . . . , v n ) ≤ C for every sequence of consecutive reflection velocities v 0 , . . . , v n and any n ≥ 0. Proof. If Q has finite escaping time, then there exists an integer K > 0 such that every billiard trajectory has at most K reflections. Since the zigzag length L : K i=1 S m → R is a continuous function with compact domain, it has a maximum. Thus, Q has bounded zigzag reflections. Now suppose that Q has not finite escaping time. This means that for every K > 0 there exists a billiard trajectory in Q that has at least K reflections with the faces of Q. By Lemma 4.2 we have v k+1 −v k ≥ δ > 0 where δ := 2 cos f (π/2)+π/2 2 > 0. This means that for every K > 0 there exists a sequence of consecutive reflection velocities v 0 , . . . , v n such that L(v 0 , . . . , v n ) ≥ δK. So Q cannot have bounded zigzag reflections.
Next we provide a sufficient condition on the contracting reflection law that guarantees boundedness of zigzag reflections. Thus finite escaping time.
Proof. Follows from (2) that Taking into account that P ηi k (u k )/ cos θ k = η i k and η i k , e = sin φ we get where h k = P η ⊥ i k (u k )/ sin θ k , e . Using classical trigonometric identities we can write To conclude the proof apply Lemma 4.2.
Proof. Let v 0 , . . . , v n be any sequence of consecutive reflection velocities. By Lemma 4.4, where To estimate γ k from below note that h k ≤ cos φ. Thus
This theorem yields the following corollaries.
Corollary 5. Any polyhedral cone Q with contracting reflection law f sufficiently close to the specular one has finite escaping time.
Proof. It is clear that 2φ > π/2 − f (π/2) for every contraction f sufficiently close to the identity. Thus, Q has finite escaping time, by Theorem 4.5.
Recall that a convex polyhedral cone Q is obtuse if φ > π/4. Corollary 6. Any obtuse polyhedral cone Q has finite escaping time for every contracting reflection law f .
5. Uniform expansion. By Proposition 3, the first component of the derivative DΦ f,P (p, v) of the billiard map is represented in Jacobi coordinates by the map where v , v, η ∈ R d are three coplanar unit vectors with v = C η (R η (v)).
The main result of this section is Theorem 5.3, which gives conditions that ensure the uniform expansion of compositions of such maps. Since the second component of the billiard map is contracting (see Proposition 4), these conditions will imply the uniform hyperbolicity of the billiard map. 5.1. Trajectories. Let P be a d-dimensional polytope in R d , and N P be the set of its unit inward normals. Denote by N 0 the set of natural numbers N including 0.
In the sequel we introduce a space of trajectories containing true orbits of the billiard map of P . The reason is to exploit the compactness of this space which does not hold for the billiard map's phase space, since one has to exclude from the phase space all orbits which eventually hit the skeleton of P .
Define the map h : Recall that D is maximal invariant set defined in Section 2.5. This map semi-conjugates the billiard map Φ f,P with the shift on the space of sequences (S × N P ) N0 . Since h(D) is not compact we introduce the following definition extending the notion of billiard trajectory.
Although N P = {η 1 , . . . , η N }, in order to simplify our notation from now on we will write η j , j ∈ N 0 , for any normal in N P and not necessarily the j-th normal in N P .
where R η is the reflection introduced in section 2, and C η is the contracting reflection law defined in subsection 2.3. We denote by T = T f,P the space of all trajectories.
Note that When the trajectory is associated to a billiard orbit {(p l , v l )} l≥0 of Φ f,P , the linear map L [i,j] represents, in Jacobi coordinates, the first component of the derivative We now extend Definition 2.8 to trajectories.
Given k ∈ N, we say that the trajectory is k-generating if it is generating on any interval [i, j] with j − i ≥ k.
We can now state this section's main result.
Theorem 5.3. Given ε > 0, d-dimensional polytope P and contracting reflection law f , there exists a constant σ = σ(ε, d, f ) > 1 such that for any trajectory {(v j , η j )} j≥0 in T f,P the following holds. If 1. P is ε-spanning, The proof of this theorem is done at the end of the section. Remark 1. From the previous theorem's conclusion, for any n ≥ 0, This means, minimum growth expansion rate larger or equal than σ 1 2k > 1.

Properties of trajectories.
The following result says that the trajectory space T is compact.
Proposition 7. The space T is a closed subspace of the product space (S × N P ) N0 . In particular, with the induced topology T is a compact space.
Proof. The trajectory space T is closed in the product space because conditions (1) and (2) in Definition 5.1 are closed conditions. By Thychonoff's theorem (S×N P ) N0 is compact, and hence T is compact too.

5.3.
Collinearities. Throughout the rest of this section, we assume that ε > 0 is fixed and that P is ε-spanning.
Consider a trajectory {(v l , η l )} l≥0 in T. For instance, if v i ∈ η i then {i} is a minimal collinearity of length 0.
Proof. The proof goes by induction on the length r = j − i. If the length is 0 then i = j and we have necessarily v i ∈ η i , in which case it is obvious that [i, i] = {i} is a collinearity. Assume now that the statement holds for all time intervals of length less than r, and let v i = λ i η i + · · · + λ j η j with j − i = r. We consider two cases: First suppose that λ i = 0. By item (1) of Lemma 5.5, Conversely, because λ i = 0 we have η i ∈ v i + N [i+1,j] which proves that where in the last equality we have used again item (1) of Lemma 5.5. Therefore, [i, j] is a collinearity in this case. Assume next that λ i = 0. By Lemma 5.4, there are scalars α i+1 and β i+1 such that v i+1 = α i+1 η i+1 + β i+1 v i . We may assume that β i+1 = 0. Otherwise v i+1 ∈ η i+1 and [i + 1, j] is a collinearity. Thus In this case Proposition 9. Given a trajectory {(v l , η l )} l≥0 and i < j ≤ j the following holds: j] . Then by Lemma 5.5, Proof. This corollary is a reformulation of item (2) of Proposition 9.

Quantifying collinearities.
We are now going to prove quantified versions of Propositions 8, 9 and Corollary 7. The following abstract continuity lemma will be useful.
Lemma 5.8. Let X be a compact topological space and f, g : X → R be continuous functions such that g(x) = 0 for all x ∈ X with f (x) = 0. Given δ > 0 there is δ > 0 such that for all x ∈ X, if f (x) < δ then g(x) < δ.
Proof. Assume, to get a contradiction, that the claimed statement does not hold. Then there is δ > 0 such that for all n ∈ N there is a point x n ∈ X with f (x n ) < 1 n and g(x n ) ≥ δ. Since X is compact, by taking a subsequence we can assume x n → x in X. By continuity of f and g, f (x) = 0 and g(x) ≥ δ, which contradicts the lemma hypothesis.
Definition 5.9. Given δ > 0, we call δ-collinearity of a trajectory {(v l , η l )} l to any Proposition 10. Given δ > 0 there exists δ > 0 such that for any trajectory {(v l , η l )} l the following holds. If is a δ-collinearity of the given trajectory.
Proof. Notice that, because the space of trajectories T is shift invariant, there is no loss of generality in assuming that [i, j] = [0, p]. For each k ≥ 0, define the functions f k , g k : T → R by These functions are clearly continuous. Proposition 8 shows that for all x ∈ T and 0 ≤ k ≤ p, f k (x) = 0 implies g k (x) = 0. Thus, given δ > 0, by Lemma 5.8, there exists δ > 0 such that for any 0 ≤ k ≤ p and x ∈ T, Proposition 11. Given any trajectory {(v l , η l )} l , i < j ≤ j and δ > 0 the following holds.

Figure 2. Composition of the projections
To prove item (2) note that, as in the proof of Proposition 10, there is no loss of generality in assuming that [i, j] = [0, p]. Define the functions f k , g k : T → R by These functions are clearly continuous.
Corollary 8. Given δ > 0 there is δ > 0 such that the following dichotomy holds. Let [i + 1, j] be a time segment of a trajectory that contains no subinterval which is a δ-collinearity of that trajectory. Then for every l ∈ [i + 1, j] either Proof. This corollary is a reformulation of Proposition 11 (2).
The remaining lemmas are abstract. Let V , V , V be Euclidean spaces of the same dimension, and L : V → V , L : V → V be linear isomorphisms.
Given σ ≥ 1 and a subspace E ⊂ V , we say that L is a σ-expansion on E if Lv ≥ σ v for all v ∈ E, i.e., m(L| E ) ≥ σ. Given another linear subspace H ⊆ V such that E ⊆ H we say that L is a relative σ-expansion on H w.r.t. E if and only if the quotient map L : V /E → V /L(E) is a σ-expansion on H/E. Note that the quotient space V /E is an Euclidean space which can naturally be identified with E ⊥ . Finally, we say that L is a σ-expansion to mean that L is a σ-expansion on its domain V .
If we do not need to specify the minimal rate of expansion we shall simply say that L is a uniform expansion on E, or that L is a relative uniform expansion on H w.r.t. E. Proof. Follows immediately from the definition of σ-expansion and relative σ -expansion.
We will now derive some explicit formulas for the minimum expansion of compositions of linear expanding maps. For that purpose we introduce an exotic operation on the set [0, 1] that plays a key role in these formulas.
The following lemmas use the notation introduced in subsection 2.1.
Proof. We can assume that To simplify the geometry we assume from now on that L =L.
The next lemma is designed to be applied to a sequence of linear maps L vi−1,ηi,vi : v ⊥ i−1 → v ⊥ i associated with an orbit segment of the billiard map Φ f,P . Compare assumptions (1)-(2) of Lemma 5.14 with the conclusions of Lemma 5.10 and Remark 2.
Proof. For each i = 1, . . . , n define, and every vector u ∈ W ⊥ i is fixed by all L j with 0 ≤ j ≤ i, we have L (i) u = u for every u ∈ W ⊥ i . We can delete from {v 0 , v 1 , . . . , v n } all vectors v i such that {v i−1 , v i } is linearly dependent, which by item (1) correspond to maps L i = id, and in this way assume that for all i = 1, . . . , n, the vectors {v i−1 , v i } are linearly independent and L i ≥ λ.
Because W ⊥ i is a singular subspace it follows that where k i := dim(W i ) − 1 and The proof of this claim goes by induction in i, applying Lemma 5.13.
The claim holds for i = 1 with k 1 = 1 and σ 1 = λ. Assume now (induction hypothesis) that To apply Proposition 5.13 we need to check that Let v 0 i denote the unit vector obtained normalizing the orthogonal projection of i . Take any unit vector v ∈ v ⊥ i−1 ∩ W i−1 and let us prove that sin(∠(v 0 i , v)) ≥ ε. This will establish (5).
which is a unit vector in W i−1 . We can assume that v 0 i , v ≥ 0 for otherwise the angle ∠(v 0 i , v) that we want to minimize would be obtuse. Using the previous expressions for v i and v we have Since this expresses v i , v as a convex combination between v 0 i , v and the number 1, it follows that This proves (5) and shows the assumptions of Proposition 5.13 are met. From this proposition, we get that on the linear subspace 5.6. Proof of Theorem 5.3. In this subsection we relate collinearities with expansion of the velocity tangent flow, and then prove Theorem 5.3.
Recall that we are assuming that P is ε-spanning.
Proposition 12. There exists σ > 1, depending only on d, f and ε, such that given a collinearity [i, j 0 ] of some trajectory, for all j > j 0 , the velocity flow , for all j > j 0 , and by Lemma 5.4, we have v j = α j η j + β j v j−1 with α j ≥ cos( π 2 λ(f )) > 0. Hence there is some 0 < ε < ε depending on ε and on λ(f ), such that for all j > j 0 with Consider the set of 'new normal' times and the corresponding velocity subspace By Lemma 5.14 there exists σ > 1, depending only on d, f and ε such that Hence there exists 1 <σ < σ depending only onε and σ such that Corollary 9. Given the constant σ > 1 in Proposition 12, and 1 < σ < σ, there is δ > 0 such that for every trajectory {(v l , η l )} l , if [i, j 0 ] is a δ-collinearity then for all j > j 0 , the velocity flow Proof. This follows from Proposition 12 with a continuity argument like the one used in the proof of Proposition 10.
Proposition 13. Given δ > 0 there exists σ > 1, depending on d, f , ε and δ, such that if a time interval [i + 1, j] of some trajectory contains no subinterval which is a δ-collinearity then Proof. Let [i, j] be a time interval such that [i+1, j] contains no subinterval which is itself a δ-collinearity. By Corollary 8, there is δ > 0 such that for every l ∈ [i + 1, j] either η l ∈ {η i+1 , . . . , η l−1 }, or else Thus by Lemma 5.14 L Now we can prove Theorem 5.3.
Fix some integer k ≥ 0 and let {(v j , η j )} j∈N0 be a trajectory. We consider three cases: Finally we consider the case [0, k] contains δ-collinearities, but the minimal ones have length zero, say {i} ⊂ [0, k] is a δ-collinearity. In this case we have ∠(v i , η i ) < δ, and the proof is somehow simpler. By Lemma 2.   Here we consider the situation when λ(f ) ≈ 0 for a given contracting reflection law f . These reflection laws are called strongly contracting (see [5]). In this context the dynamics may loose uniformity due to unbounded escaping times. To any strongly contracting billiard we can associate a degenerate billiard map called the 'slap map' corresponding to f = 0, where reflections are always orthogonal to the faces. When h is small enough the slap map has a trapping region, called a chamber, away from acute wedges. Hence the escaping time is bounded on the chamber. This concept generalizes the notion of chamber introduced in [6].
For simplicity we will assume d = 3.
Proof. Firstly, let us assume that λ(f ) = 0. This means that the billiard particle always reflects orthogonally to each face of the polytope. Since after the first iterate the angle is zero, we can reduce Φ f,∆ 3 h to a multi-valued map Φ 0 : ∆ 3 h → ∆ 3 h (skeleton points may have more than one image). Let A i , i = 1, . . . , 4 denote the vertexes of the simplex ∆ 3 h with A 4 being the top vertex. The triangle A 1 A 2 A 3 is called the base of the simplex (see Figure 4). We show that there is a set V on the base of the simplex which is invariant by Φ 2 0 . Let C 0 denote the center of A 1 A 2 A 3 , i.e., the point mapped by Φ 0 to the top vertex of the simplex. Then, the base triangle is partitioned into three triangles, namely A 1 A 2 C 0 , A 1 A 3 C 0 and A 2 A 3 C 0 . Since Φ 0 (C 0 ) is the intersection of the three faces, it has three distinct images by Φ 0 . A simple calculation shows that when h < h 0 for some h 0 > 0, these images belong to the base of the simplex. Denote them by C 1 , C 2 and C 3 . The image of triangles A 1 A 2 C 0 , A 2 A 3 C 0 and A 3 A 1 C 0 under Φ 2 0 are respectively triangles A 1 A 2 C 3 , A 2 A 3 C 1 and A 3 A 1 C 2 . Therefore Φ 2 0 maps the triangle A 1 A 2 A 3 to itself. Now we construct an hexagon H = M 1 M 2 M 3 M 4 M 5 M 6 as follows (see Figure 4): the point M 1 is the intersection of A 1 C 2 with the perpendicular to A 1 C 0 through C 1 . Likewise, M 2 is the intersection of A 2 C 1 with the perpendicular to A 2 C 0 through C 2 . The other M j 's are similarly defined. The hexagon H is the union of three pentagons whose images under Φ 2 0 are in the hexagon H. On the right of Figure 4, we can see the image of the pentagon P = C 0 C 2 M 2 M 1 C 1 , Φ 2 0 (P) = C 0 C 2 M 2 M 1 C 1 . Moreover, the intersection of the pentagon Φ 2 0 (P) with the boundary of H is just the point C 0 = C 3 . Hence, for some small enough neighborhood V of H on the base triangle A 1 A 2 A 3 we have Φ 2 0 (V) ⊂ V. It is easy to see that every orbit of Φ 0 eventually enters V. In fact, every orbit starting near the wedges of the simplex will escape by a zig-zag movement and enter H ⊂ V. Since V is away from the wedges, the escaping time T (x) for x ∈ V is uniformly bounded.
Also define Λ λ0 := V × S + η,λ0 ∩ M + . Then, by continuity, there exists λ 0 = λ 0 (h) > 0 such that for every contracting reflection law satisfying λ(f ) < λ 0 we have The previous equality follows from the fact that for each x ∈ M + there exists n ≥ 0 such that Φ n f,∆ 3 h (x) ∈ Λ λ0 . Since the escaping time is bounded on Λ λ0 , it is also bounded on D. Therefore, the proposition follows from Theorem 2.13. CEMAPRE -UID/MULTI/00491/2013 financed by FCT/MEC through national funds. MS was supported by PNPD/CAPES. The authors wish to express their gratitude to Gianluigi Del Magno for stimulating conversations and also to the anonymous referee that helped us to significantly improve the presentation of the paper.