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On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point

Claudio Bonanno, Sharvari Neetin Tikekar

TL;DR

The paper addresses the escape rate for intermittent (parabolic) interval maps with holes shrinking to a parabolic fixed point. It develops a transfer-operator and inducing framework that links the escape rate of the original parabolic map to that of its induced jump map, enabling exact relations and sharp asymptotics. For Markov holes, the authors prove an explicit formula relating the two escape rates and show that, as the hole size $m(H)$ tends to zero, the escape rate scales as $m(H)^s$ when $s>1$, as $m(H)/(-\log m(H))$ when $s=1$, and linearly with $m(H)$ when $s\in(0,1)$. These results extend the open dynamical systems theory to non-uniformly expanding maps with indifferent fixed points and potentially infinite invariant measures, providing a general method and precise bounds for shrinking holes.

Abstract

We study non-uniformly expanding maps of the unit interval with a parabolic fixed point at the origin that admit an ergodic absolutely continuous invariant measure, which may be finite or infinite. By introducing a hole defined by an interval containing the parabolic fixed point, we analyze the escape rate of the resulting open system and its asymptotic behavior as the hole shrinks. Our approach relies on the transfer operator associated with the dynamical system and on the relationship between the transfer operators of the original system and its induced version. The results extend to this general framework previous investigations which considered special cases.

On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point

TL;DR

The paper addresses the escape rate for intermittent (parabolic) interval maps with holes shrinking to a parabolic fixed point. It develops a transfer-operator and inducing framework that links the escape rate of the original parabolic map to that of its induced jump map, enabling exact relations and sharp asymptotics. For Markov holes, the authors prove an explicit formula relating the two escape rates and show that, as the hole size tends to zero, the escape rate scales as when , as when , and linearly with when . These results extend the open dynamical systems theory to non-uniformly expanding maps with indifferent fixed points and potentially infinite invariant measures, providing a general method and precise bounds for shrinking holes.

Abstract

We study non-uniformly expanding maps of the unit interval with a parabolic fixed point at the origin that admit an ergodic absolutely continuous invariant measure, which may be finite or infinite. By introducing a hole defined by an interval containing the parabolic fixed point, we analyze the escape rate of the resulting open system and its asymptotic behavior as the hole shrinks. Our approach relies on the transfer operator associated with the dynamical system and on the relationship between the transfer operators of the original system and its induced version. The results extend to this general framework previous investigations which considered special cases.
Paper Structure (5 sections, 6 theorems, 83 equations)

This paper contains 5 sections, 6 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The maps
  3. The symbolic representation
  4. Transfer operators
  5. The open systems and the escape rate

Key Result

Theorem 2.2

Assuming (V1)-(V2) and (W1)-(W2), for $|z|\le 1$ and ${\mathcal{M}}_z$ acting on ${\mathcal{H}}_{\theta}(\Sigma)$, the following hold: Let us now consider the special case $z=1$ and let ${\mathcal{M}}:= {\mathcal{M}}_1$.Then, $1$ is the largest simple eigenvalue of ${\mathcal{M}}$ with positive eigenfunction $h \in {\mathcal{H}}_{\theta}(\Sigma)$ bounded from above and away from 0. The dual opera

Theorems & Definitions (14)

  • Example 2.1
  • Theorem 2.2: Ruelle-Perron-Frobenius Theorem for ${\mathcal{M}}_z$ Is
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 4 more