## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
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- Mathematics in Engineering

### Open Access Journals

JMD

We study the billiard map associated with both the finite- and
infinite-horizon Lorentz gases having smooth scatterers with strictly
positive curvature. We introduce generalized function spaces (Banach spaces
of distributions) on which the transfer operator is quasicompact. The
mixing properties of the billiard map then imply the existence of a
spectral gap and related statistical properties such as exponential decay
of correlations and the Central Limit Theorem. Finer statistical properties
of the map such as the identification of Ruelle resonances, large deviation
estimates and an almost-sure invariance principle follow immediately once
the spectral picture is established.

DCDS

In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.

keywords:
nonuniformly hyperbolic
,
flat point.
,
infinite horizon
,
Dispersing billiards
,
free path

## Year of publication

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