The Lyapunov exponent in the Sinai billiard in the small scatterer limit

Abstract

We show that Lyapunov exponent for the Sinai billiard is $\lambda =-2\log (R)+C+O(R{\log }^{2}R)$ with $C=1-4\log 2+27/(2{\pi }^2)\cdot \zeta (3)$ where $R$ is the radius of the circular scatterer. We consider the disk-to-disk-map of the standard configuration where the disks is centered inside a unit square.

15 pages LaTeX, 3 (useful) figures available from the author

References in corpus (4)

• Lyapunov Exponents and KS Entropy for the Lorentz Gas at Low Densities
• On the decay of correlations in Sinai billiards with infinite horizon
• Approximate zeta functions for the Sinai billiard and related systems
• Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

Cited by in corpus (22)

• Anomalous transport: a deterministic approach
• The statistics of the trajectory in a certain billiard in a flat two-torus
• On the decay of correlations in Sinai billiards with infinite horizon
• Kinetic transport in the two-dimensional periodic Lorentz gas
• Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases I: Equilibrium Systems
• Free path lengths in quasicrystals
• Recent Results on the Periodic Lorentz Gas
• Periodic orbit theory of strongly anomalous transport
• The average length of a trajectory in a certain billiard in a flat two-torus
• Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes
• Power-law distributions for the free path length in Lorentz gases
• The low-density limit of the Lorentz gas: periodic, aperiodic and random
• Transmission and Scattering of a Lorentz Gas on a Slab
• Universal hitting time statistics for integrable flows
• The role of singularities in chaotic spectroscopy
• Distribution of the linear flow length in a honeycomb in the small-scatterer limit
• Error of semiclassical eigenvalues in the semiclassical limit - An asymptotic analysis of the Sinai billiard
• Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes
• On the distribution of the free path length of the linear flow in a honeycomb
• Kinetic Theory for the Low-Density Lorentz Gas
• The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit
• The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates