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Return Times Distribution of Expanding Maps

Nicolai T A Haydn

TL;DR

This work establishes that for expanding maps with near-uniform expansion except on a small zero-measure set, the limiting statistics of hitting times to shrinking neighborhoods of that set are compound Poisson, provided cluster-size limits exist and correlations decay at least polynomially. The authors develop a general blocking/compound-binomial framework and verify it under a set of geometric, distortion, and dimension assumptions, yielding explicit conditions under which Poisson or Polya-Aeppli limits arise. They apply the theory to C^2 interval maps, product maps, and parabolic interval maps, deriving both standard and nonstandard (parabolic) scalings and demonstrating how the extremal index and Pitskel values govern the limiting cluster structure. The results extend prior pointwise targets to zero-measure target sets and offer a unified approach to return-time statistics in a broad class of expanding and near-parabolic dynamical systems, with implications for extremal processes and statistical physics models. All formulas are presented with explicit scaling and limit relations, facilitating application to concrete systems and further generalizations.

Abstract

We consider expanding systems with invariant measures that are uniformly expanding everywhere except on a small measure set and show that the limiting statistics of hitting times for zero measure sets are compound Poisson provided the limits for the cluster size distributions exist. This extends previous results from neighbourhoods around single points to neighbourhoods around zero measure sets. The assumptions require the correlations to decay at least polynomially and the non-uniformly expanding part of the iterates of the map also has to satisfy some decay condition. We also require some regularity conditions around the limiting zero measure target set.

Return Times Distribution of Expanding Maps

TL;DR

This work establishes that for expanding maps with near-uniform expansion except on a small zero-measure set, the limiting statistics of hitting times to shrinking neighborhoods of that set are compound Poisson, provided cluster-size limits exist and correlations decay at least polynomially. The authors develop a general blocking/compound-binomial framework and verify it under a set of geometric, distortion, and dimension assumptions, yielding explicit conditions under which Poisson or Polya-Aeppli limits arise. They apply the theory to C^2 interval maps, product maps, and parabolic interval maps, deriving both standard and nonstandard (parabolic) scalings and demonstrating how the extremal index and Pitskel values govern the limiting cluster structure. The results extend prior pointwise targets to zero-measure target sets and offer a unified approach to return-time statistics in a broad class of expanding and near-parabolic dynamical systems, with implications for extremal processes and statistical physics models. All formulas are presented with explicit scaling and limit relations, facilitating application to concrete systems and further generalizations.

Abstract

We consider expanding systems with invariant measures that are uniformly expanding everywhere except on a small measure set and show that the limiting statistics of hitting times for zero measure sets are compound Poisson provided the limits for the cluster size distributions exist. This extends previous results from neighbourhoods around single points to neighbourhoods around zero measure sets. The assumptions require the correlations to decay at least polynomially and the non-uniformly expanding part of the iterates of the map also has to satisfy some decay condition. We also require some regularity conditions around the limiting zero measure target set.
Paper Structure (17 sections, 10 theorems, 124 equations)

This paper contains 17 sections, 10 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Expanding Maps and Limiting Distributions
  3. Assumptions and main result
  4. Return times and Kac's scaling
  5. The Compound Binomial Approximation
  6. Proof of Theorem \ref{['main.theorem']}
  7. Compound binomial approximation of the return times distribution
  8. Estimating $\mathcal{R}_1$
  9. Estimating the terms $\mathcal{R}_2$
  10. The total error
  11. Convergence to the compound Poisson distribution
  12. Examples
  13. $C^2$ interval maps
  14. Product of interval maps
  15. Parabolic interval maps
  16. ...and 2 more sections

Key Result

Theorem 3.1

Assume that the map $T: \Omega\to \Omega$ satisfies the assumptions (I)--(IV) where $\mathcal{C}(k)$ decays at least polynomially with power $\mathfrak{p}>(\frac{\beta}{\eta}+d_1)\frac{1+\mathfrak{d}}{d_0}$ and $\mathfrak{k}>\frac{1+\mathfrak{d}}{d_0}$. Let $\Gamma\subset \Omega$ be a zero measure s where $\nu$ is the compound Poisson distribution for the parameters $t$, $\lambda_\ell$ and $N=N(L,

Theorems & Definitions (14)

  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Remark 4.1
  • Theorem 5.1
  • Theorem 7.1
  • Corollary
  • Theorem 7.2
  • Lemma 7.1
  • ...and 4 more