[Paper Review] Statistics of closest returns for some non-uniformly hyperbolic systems

Pierre Collet|ArXiv.org|Apr 23, 1999

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper establishes that for non-uniformly hyperbolic interval maps with exponential decay of correlations, the distribution of closest returns to a given point, when properly normalized, converges almost surely to an exponential law. A similar result holds when tracking returns to the initial point of the trajectory, demonstrating universal statistical behavior in such systems under mild mixing conditions.

ABSTRACT

For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non uniformly hyperbolic dynamical systems developed by L.S.Young.

Motivation & Objective

• To understand the statistical distribution of closest returns in non-uniformly hyperbolic dynamical systems.
• To determine whether the normalized time to the first return near a given point converges to an exponential distribution.
• To extend results on return times to systems that are not uniformly hyperbolic but exhibit exponential decay of correlations.
• To analyze the case where the reference point is the initial point of the trajectory, not a fixed point in phase space.
• To establish almost sure asymptotic convergence to an exponential law under suitable normalization.

Proposed method

• The analysis relies on the framework of non-uniformly hyperbolic systems developed by the author and collaborators.
• It uses the exponential decay of correlations as a key assumption to control mixing behavior.
• The authors normalize the return times by a function scaling with the system's mixing rate and the point's recurrence properties.
• The proof employs probabilistic techniques to show convergence in distribution and almost sure convergence of the normalized return times.
• The method applies to interval maps with a specific class of non-uniform hyperbolicity, such as certain unimodal maps.
• The normalization accounts for the local recurrence rate and ensures convergence to the standard exponential distribution.

Experimental results

Research questions

• RQ1 Does the distribution of closest return times to a fixed point in a non-uniformly hyperbolic system converge to an exponential law when properly normalized?
• RQ2 How does the statistical behavior of return times change when the reference point is the initial point of the trajectory rather than a fixed point?
• RQ3 To what extent does exponential decay of correlations ensure universal asymptotic exponential statistics for return times?
• RQ4 Can the asymptotic exponential law be established almost surely, not just in distribution, for such systems?
• RQ5 What normalization is required to achieve convergence to the standard exponential distribution in this setting?

Key findings

• The normalized closest return time to a given point converges almost surely to an exponential distribution.
• The same asymptotic exponential law holds when the reference point is the initial point of the trajectory.
• The result is established under the assumption of exponential decay of correlations in non-uniformly hyperbolic interval maps.
• The normalization used depends on the system's mixing rate and the recurrence properties of the point.
• The convergence is both in distribution and almost surely, indicating strong statistical regularity.
• The findings extend the classical Poisson-type statistics of return times to a broader class of non-uniformly hyperbolic systems.