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Averaging theorems for slow fast systems in $\mathbb{Z}$-extensions (discrete time)

Maxence Phalempin

Abstract

We study the averaging method for flows perturbed by a dynamical system preserving an infinite measure. Motivated by the case of perturbation by the collision dynamic on the finite horizon $\mathbb Z$-periodic Lorentz gas and in view of future development, we establish our results in a general context of perturbation by $\mathbb Z$-extension over chaotic probability preserving dynamical systems. As a by product, we prove limit theorems for non-stationary Birkhoff sums for such infinite measure preserving dynamical systems.

Averaging theorems for slow fast systems in $\mathbb{Z}$-extensions (discrete time)

Abstract

We study the averaging method for flows perturbed by a dynamical system preserving an infinite measure. Motivated by the case of perturbation by the collision dynamic on the finite horizon -periodic Lorentz gas and in view of future development, we establish our results in a general context of perturbation by -extension over chaotic probability preserving dynamical systems. As a by product, we prove limit theorems for non-stationary Birkhoff sums for such infinite measure preserving dynamical systems.
Paper Structure (11 sections, 17 theorems, 164 equations, 1 figure)

This paper contains 11 sections, 17 theorems, 164 equations, 1 figure.

Table of Contents

  1. introduction
  2. Collision dynamics associated to $\mathbb{Z}$-periodic Lorentz gas with finite horizon.
  3. General results for $\mathbb Z$-extensions
  4. General model of $\mathbb{Z}$-extension.
  5. Perturbed ergodic Birkhoff sum, result for transformation
  6. Slow-fast systems (perturbation by the map)
  7. Perturbed ergodic sums : proof of the limit theorem.
  8. Convergence in law.
  9. Proof of theorem \ref{['thmbirkhoff']} : finite dimensional distributions.
  10. Proof of theorem \ref{['thmbirkhoff']}: tightness.
  11. Averaging : Proof of theorem \ref{['thmequadif']}.

Key Result

Theorem 1.1

Let $\epsilon >0$, and $(M,T,\mu)$ be the collision dynamics associated to the $\mathbb{Z}$-periodic Lorentz gas with finite horizon (see Section secmodels) and suppose $F'(x,\omega)=\overline F(x)+F(x,\omega)$ with $\overline F : \mathbb R^d \rightarrow \mathbb R^d$ and $F: \mathbb R^d \times M \ri where $\mathcal{L}_\mu$ denotes the strong convergence in lawThis means the convergence in distribu

Figures (1)

  • Figure 1: Illustration of the map $T$ corresponding to the dynamics of the discrete time $\mathbb{Z}$-periodic Lorentz gas in finite horizon, with two periodic shape of obstacles.

Theorems & Definitions (33)

  • Theorem 1.1: See Theorem \ref{['thmequadif']} for a more general statement
  • Theorem 1.2: see Theorem \ref{['thmbirkhoff']} page \ref{['thmbirkhoff']}
  • Remark 3.3: Application to the periodic Lorentz gas
  • Theorem 3.4: Convergence of perturbed Birkhoff sums for the map
  • Remark 3.5
  • Theorem 3.6
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • ...and 23 more