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Polynomial escape rates via maximal large deviations

Yaofeng Su

TL;DR

The paper addresses open dynamical systems with intermittency and polynomial escape rates, focusing on the tail behavior of $\mu(\tau_H>n)$. It proposes a concise method based on maximal large deviations (as developed in $\cite{mldp}$) applied to Young towers with polynomial mixing to derive polynomial escape rates. For one-dimensional Liverani–Saussol–Vaienti maps and for solenoids with intermittency, it shows that, under mild hole conditions, the escape rate matches the mixing rate, e.g., $\mu(\tau_H>n)\asymp n^{1-1/\alpha}$ when the mixing rate is $O(n^{1-1/\alpha})$. The framework provides a compact, adaptable approach for establishing polynomial escape rates across a range of intermittent open systems, extending to higher-dimensional hyperbolic dynamics without requiring Markov holes.

Abstract

In this short note, we propose a new and short approach to polynomial escape rates, which can be applied to various open systems with intermittency. The tool of our approach is the maximal large deviations developed in \cite{mldp}.

Polynomial escape rates via maximal large deviations

TL;DR

The paper addresses open dynamical systems with intermittency and polynomial escape rates, focusing on the tail behavior of . It proposes a concise method based on maximal large deviations (as developed in ) applied to Young towers with polynomial mixing to derive polynomial escape rates. For one-dimensional Liverani–Saussol–Vaienti maps and for solenoids with intermittency, it shows that, under mild hole conditions, the escape rate matches the mixing rate, e.g., when the mixing rate is . The framework provides a compact, adaptable approach for establishing polynomial escape rates across a range of intermittent open systems, extending to higher-dimensional hyperbolic dynamics without requiring Markov holes.

Abstract

In this short note, we propose a new and short approach to polynomial escape rates, which can be applied to various open systems with intermittency. The tool of our approach is the maximal large deviations developed in \cite{mldp}.
Paper Structure (4 sections, 2 theorems, 10 equations)

This paper contains 4 sections, 2 theorems, 10 equations.

Table of Contents

  1. introduction
  2. Definitions and notations
  3. One-dimensional intermittent maps
  4. Solenoids with intermittency

Key Result

theorem 3.1

For the dynamical system $(X,f,\mu)$:= $([0,1],g_{\alpha}, \mu_{\alpha})$, if a (dis)connected hole $H$ satisfies $0\notin cls(H)$ and $int(H)\neq \emptyset$, then $\mu(\tau_H>n)\approx n^{1-1/\alpha}$.

Theorems & Definitions (9)

  • theorem 3.1: Escape Rates I
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 4.1: Escape Rates II
  • proof
  • remark thmcounterremark
  • remark thmcounterremark