Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows

Abstract

We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group L_t of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions. (Revised after referees' comments.)

v2: Revised version after referees' comments. (The homogeneity layers do not appear anymore in the def. of the norm.) The text published in Invent. Math. is v2, but has more pages (due to smaller margins in v2). v3: Contains an additional 2-page note correcting flaws found by M. Jezequel (L 3.2) and D. Thomine (L 4.6 and L 7.5)

References in corpus (1)

• Exponential decay of correlations for nonuniformly hyperbolic flows with a C^{1+α} stable foliation, including the classical Lorenz attractor

Cited by in corpus (32)

• The quest for the ultimate anisotropic Banach space
• Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables
• On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps
• Superpolynomial and polynomial mixing for semiflows and flows
• A gentle introduction to anisotropic Banach spaces
• Polynomial decay of correlations for flows, including Lorentz gas examples
• Horocycle averages on closed manifolds and transfer operators
• Smooth mixing Anosov flows in dimension three are exponential mixing
• Elliptic Flowers: simply connected billiard tables with chaotic or non-chaotic flows moving around chaotic or non-chaotic cores
• Transfer operator approach to ray-tracing in circular domains
• Thermodynamic formalism for dispersing billiards
• Markov partition and Thermodynamic Formalism for Hyperbolic Systems with Singularities
• Equidistribution for standard pairs in planar dispersing billiard flows
• Discontinuities cause essential spectrum
• Projective Cones for Sequential Dispersing Billiards
• Closed geodesics with prescribed intersection numbers
• Fluctuation of the entropy production for the Lorentz gas under small external forces
• Uniform hyperbolicity in nonflat billiards
• Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas
• Mathematical Study of Scattering Resonances
• Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators
• Prime orbit theorems for expanding Thurston maps: Dirichlet series and orbifolds
• Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces
• Thermodynamic formalism for piecewise expanding maps in finite dimension
• The first encounter of two billiard particles of small radius
• Spectral theory of the frame flow on hyperbolic 3-manifolds (with an appendix by Charles Hadfield)
• Nonstationary open dynamical systems
• Thermodynamic formalism for discontinuous maps and statistical properties of their equilibrium states
• Prime orbit theorems for expanding Thurston maps
• Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps
• Sinai billiard maps with Ruelle resonances
• The spectral gap for transfer operators of torus extensions over expanding maps