Spectral analysis of hyperbolic systems with singularities

Abstract

We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.

On the measure of maximal entropy for finite horizon Sinai Billiard maps

“…It follows from [DZ3,Lemma 5.3] that 1 B n,ε ν ∈ B w . In the proof of Lemma 3.4, it was shown that if x, y lie in different elements of M n 0 , then d n (x, y) ≥ ε 0 , where d n (·, ·) is the dynamical distance defined in (2.1).…”

A gentle introduction to anisotropic banach spaces

“…[DZ3] has the further problem that (H1) of that paper is not satisfied in the present setting. (H1) would require that the Jacobian of T κ,λ , which equals κλ, is large compared to the contraction obtained in the Lasota-Yorke inequalities, i.e.…”

“…For systems with discontinuities, integrating on stable curves greatly simplifies the geometric arguments required to control the growth in complexity due to discontinuities. This was implemented first for two-dimensional piecewise hyperbolic maps (with bounded derivatives) [DL], and then for various classes of billiards [DZ1,DZ3] and their perturbations [DZ2]. It has also led to the recent proof of exponential mixing for some billiard flows [BDL].…”

Local Limit Theorem for Randomly Deforming Billiards

“…Fix β > 0. Due to [1,Lemma 4.3], Lemma C.1, and the continuity in u provided by [16,Lemma 5.4] (see also Lemma 3.16 applied to L u,0 rather than P u ), we know that there exists C > 1 and α ∈ (0, 1) such that ∀n ∈ N * , sup β≤|u|≤π L n u,0 L(B,B) ≤ Cα n .…”