[Paper Review] Recurrence, dimensions and Lyapunov exponents

Benoît Saussol, Serge Troubetzkoy+1 more|arXiv (Cornell University)|Sep 25, 2001

Mathematical Dynamics and FractalsMathematics8 references19 citations

TL;DR

This paper establishes fundamental relationships between recurrence times, Lyapunov exponents, and dimension in one-dimensional dynamical systems. It proves that for typical points, the logarithmic return time of shrinking neighborhoods scales linearly with the inverse logarithm of radius, and this scaling rate equals the local Hausdorff dimension. The key contribution is a direct link between the asymptotic behavior of return times and key dynamical invariants—Lyapunov exponent and dimension—via the thermodynamics of return times.

ABSTRACT

We show that the Poincaré return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.

Motivation & Objective

• To establish a theoretical foundation for the thermodynamics of return times in dynamical systems.
• To derive lower bounds on the first return time of cylinders in systems with positive metric entropy.
• To express Lyapunov exponents and Hausdorff dimension of invariant measures via return time statistics.
• To demonstrate that the limit of log(return time)/(-log r) as r→0 equals the local dimension μ-almost everywhere.

Proposed method

• Uses Kolmogorov complexity and White’s sharpening of Brudno’s theorem to bound return time growth rates.
• Applies the concept of cylinder sets and neighborhood return times τr(x) for points in one-dimensional maps.
• Employs Hofbauer’s comparison between balls and cylinders to relate return times to metric entropy and Lyapunov exponents.
• Establishes asymptotic bounds on lim inf and lim sup of log τr(x) / (-log r) using entropy and Lyapunov exponent estimates.
• Combines results from symbolic dynamics, measure-theoretic entropy, and dimension theory to prove convergence of return time scaling.
• Uses the fact that for typical points, the return time of a ball of radius r scales as r^(-dμ(x)) where dμ(x) is the local dimension.

Experimental results

Research questions

• RQ1 What is the minimal possible growth rate of the first return time of cylinders in a system with positive metric entropy?
• RQ2 Can the Lyapunov exponent be recovered from the asymptotic behavior of return times to shrinking neighborhoods?
• RQ3 Does the limit lim_{r→0} log τr(x) / (-log r) exist μ-almost everywhere, and if so, what does it equal?
• RQ4 How are the local dimension dμ(x), metric entropy hμ, and Lyapunov exponent λμ related through return time statistics?

Key findings

• For any ergodic system with positive metric entropy, the lower limit of τ(ζⁿˣ)/n is almost everywhere at least 1, establishing a fundamental lower bound on recurrence.
• The limit lim_{r→0} log τr(x) / (-log r) exists μ-almost everywhere and equals the local Hausdorff dimension dμ(x).
• For a large class of one-dimensional maps, the local dimension dμ(x) is equal to hμ / λμ, and this is also equal to the limit of log τr(x) / (-log r).
• The upper and lower bounds on the return time scaling are shown to converge to hμ / λμ, proving the equality of the limit and the dimension ratio.
• The result holds even for maps with critical and parabolic points, demonstrating broad applicability beyond uniformly hyperbolic systems.
• The proof relies on sharp comparisons between balls and cylinders via Hofbauer’s results and uses Kolmogorov complexity to control entropy growth.