An Expansion Estimate for Dispersing Planar Billiards with Corner Points

Abstract

Abstract. It is known that the dynamics of planar billiards satisfies strong mixing properties (e.g. exponential decay of correlations) provided that some expansion condition on unstable curves is satisfied. This condition has been shown to always hold for smooth dispersing planar billiards, but it needed to be assumed separately in the case of dispersing planar billiards with corner points.We prove that this expansion condition holds for any dispersing planar billiard with corner points, no cusps and bounded horizon.

The first encounter of two billiard particles of small radius

“…(We need to choose N such that LN 3 < Λ N so as to bound λ(W i,N ) for i's which never visited the nearly grazing domains H k with |k| ≥ k 0 , and choose k 0 big to bound the remaining terms.) Finally, the proof of Lemma 1 based on ( 8) is again similar to the usual argument (see also [7,10]).…”

“…We note that similar N-step expansions have been used several times, e.g. in case of billiards with corner points [7,10].…”

“…There is two reasons why (Φ s ) n can cut an unstable curve: grazing collisions and collisions at times which are integer multiples of s. The grazing collisions are liable for the fragmentation of unstable curves of the map F , hence can be bounded by (10). Namely, for fixed n and for small enough δ, W can be cut into pieces W α , α ≤ Bn such that all X, Y belonging to the same piece W α collide on the same sequence of scatterers during flow time ns.…”

Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

“…• Planar periodic dispersing billiards with finite horizon [49] and infinite horizon [10], as well as billiards with external forcing and corners [10,13].…”

Multidimensional hyperbolic billiards

“…The reader is reminded that for simplicity we assume that the boundaries of the scatterers are smooth, i. e. there are no corner points (they may cause additional problems, in particular in obtaining appropriate complexity bounds, cf. [44]). (1) The condition on algebraicity of ∂Q follows from a weaker technical one: there exists a constant L ∈ R + such that for every N ∈ N the singularity set ∪ n≤N S n is Lipschitz decomposable with constant L (i. e. it is a finite collection of graphs of Lipschitz functions, cf.…”