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Local large deviations for periodic infinite horizon Lorentz gases

Ian Melbourne, Francoise Pene, Dalia Terhesiu

Abstract

We prove local large deviations for the periodic infinite horizon Lorentz gas viewed as a ${\mathbb Z}^d$-cover ($d=1,2$) of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuniformly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.

Local large deviations for periodic infinite horizon Lorentz gases

Abstract

We prove local large deviations for the periodic infinite horizon Lorentz gas viewed as a -cover () of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuniformly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.

Paper Structure

This paper contains 13 sections, 28 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Set up
  4. The range $n\ll \log |N|$.
  5. Key estimates on the one-sided tower
  6. Proof of Lemmas \ref{['lem:key']} and \ref{['lem:key2']}
  7. Renewal operators
  8. Further estimates
  9. Spectral properties for ${\widehat{R}}(z,t)$
  10. Completion of the proof of Lemmas \ref{['lem:key']} and \ref{['lem:key2']}
  11. Proof of the main result
  12. LLD for nonuniformly hyperbolic systems modelled by Young towers
  13. Acknowledgements

Key Result

Theorem 1.1

There exists $C>0$ such that for all $n\ge1$, $N\in\mathbb{Z}^d$,

Theorems & Definitions (33)

  • Theorem 1.1: LLD for the dispersing billiard
  • Remark 1.2
  • Proposition 2.1
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.3
  • Corollary 4.4
  • Proposition 5.1
  • Proposition 5.2
  • ...and 23 more