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Which subsets and when orbits of non-uniformly hyperbolic systems prefer to visit: operator renewal theory approach

Leonid A. Bunimovich, Yaofeng Su

TL;DR

This work develops an operator renewal framework to study finite-time transport in slowly mixing non-uniformly hyperbolic systems, focusing on how orbits visit subsets of the phase space and when. By constructing open-operator renewal equations and extending Keller–Liverani perturbation theory to this setting, the authors derive a detailed polynomial-decay expansion for the first-hitting/escape probabilities, revealing a leading term shared across small holes and a hole-dependent secondary term. The approach yields not only asymptotics but also finite-time visitation predictions, with explicit formulas connecting visit likelihood to hole position via constants c_H and the tail behavior of return times. Applications to one-dimensional models, including Liverani–Saussol–Vaienti, Farey, and intermittent maps, illustrate how the method can quantify finite-time transport and visitation preferences in slowly mixing dynamics. The results advance finite-time dynamical predictions and provide a robust spectral-analytic toolkit for open systems in slow-matching regimes, with potential extensions to higher-dimensional and semi-flow settings.

Abstract

The paper addresses for the first time some basic questions in the theory of finite time dynamics and finite time predictions for slowly mixing non-uniformly hyperbolic dynamical systems. It is concerned with transport in phase spaces of such systems, and analyzes which subsets and when the orbits prefer to visit. An asymptotic expansion of the decay of polynomial escape rates is obtained, which also allows for finding asymptotics of the first hitting probabilities. Our approach is based on the construction of new operator renewal equations for open dynamical systems and on their spectral analysis. In order to do this, we generalize the Keller-Liverani perturbation technique. Applications to a large class of one-dimensional non-uniformly expanding systems are considered.

Which subsets and when orbits of non-uniformly hyperbolic systems prefer to visit: operator renewal theory approach

TL;DR

This work develops an operator renewal framework to study finite-time transport in slowly mixing non-uniformly hyperbolic systems, focusing on how orbits visit subsets of the phase space and when. By constructing open-operator renewal equations and extending Keller–Liverani perturbation theory to this setting, the authors derive a detailed polynomial-decay expansion for the first-hitting/escape probabilities, revealing a leading term shared across small holes and a hole-dependent secondary term. The approach yields not only asymptotics but also finite-time visitation predictions, with explicit formulas connecting visit likelihood to hole position via constants c_H and the tail behavior of return times. Applications to one-dimensional models, including Liverani–Saussol–Vaienti, Farey, and intermittent maps, illustrate how the method can quantify finite-time transport and visitation preferences in slowly mixing dynamics. The results advance finite-time dynamical predictions and provide a robust spectral-analytic toolkit for open systems in slow-matching regimes, with potential extensions to higher-dimensional and semi-flow settings.

Abstract

The paper addresses for the first time some basic questions in the theory of finite time dynamics and finite time predictions for slowly mixing non-uniformly hyperbolic dynamical systems. It is concerned with transport in phase spaces of such systems, and analyzes which subsets and when the orbits prefer to visit. An asymptotic expansion of the decay of polynomial escape rates is obtained, which also allows for finding asymptotics of the first hitting probabilities. Our approach is based on the construction of new operator renewal equations for open dynamical systems and on their spectral analysis. In order to do this, we generalize the Keller-Liverani perturbation technique. Applications to a large class of one-dimensional non-uniformly expanding systems are considered.

Paper Structure

This paper contains 25 sections, 30 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Main results
  4. Preliminary results
  5. Several (complex) operators
  6. Uniform Lasota-Yorke inequalities
  7. Regularities of $\mathring{R}(\cdot)$ and $R(\cdot)$
  8. Spectral analysis of $[I-\mathring{R}(z)]^{-1}$ for any $z \in \overline{\mathbb{D}}$ and a small hole $H$
  9. Scheme of the proof
  10. Proof of Theorem \ref{['thm']}
  11. $\mu_{\Delta}(\tau_H>n)$ in terms of operators
  12. Partitions of unity for $[I-\mathring{R}(\cdot)]^{-1}$ on $S^1$
  13. Approximations for $\phi(\cdot)[I-\mathring{R}(\cdot)]^{-1}$ on $S^1$
  14. Approximations for $\frac{1}{1-\mathring{\lambda}(\cdot)}\phi(\cdot)\mathring{\mathop{\mathrm{Proj}}\nolimits}(\cdot)$ on $S^1$
  15. Approximations for $\frac{1}{1-\mathring{\lambda}(1)}\mathring{\mathop{\mathrm{Proj}}\nolimits}(1)$
  16. ...and 10 more sections

Key Result

theorem 1

Consider the Young tower ($\Delta, F, \mu_{\Delta}$) in Definition defyoung with decay of the tail of order $n^{-k-\beta}$. Let the center of the hole H be $z_0\in S^c \bigcap X$ (see Definition defopen) and $\mu_X=\frac{\mu_{\Delta}|_X}{\mu_{\Delta}(X)}$, then there is a small $\sigma>0$ such that where the constants in $o(\cdot),O(\cdot)$ depend on $\sigma, z_0$ but do not depend on $\mu_X(H),\

Theorems & Definitions (78)

  • definition thmcounterdefinition: Polynomial Young towers
  • remark thmcounterremark
  • definition thmcounterdefinition: Young towers with holes, escape times and hitting times
  • theorem 1: Asymptotic expansions for polynomial escape rates
  • corollary thmcountercorollary: Where orbits prefer to visit in $\Delta$
  • corollary thmcountercorollary: Finite time predictions
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • ...and 68 more