Normal distribution of Lyapunov exponents of periodic orbits for expanding circle maps

TL;DR

This paper proves that for smooth expanding circle maps, the distribution of Lyapunov exponents of periodic points approximates a normal distribution as the period increases, demonstrating a CLT-like behavior.

Contribution

It establishes a Central Limit Theorem for the empirical distribution of Lyapunov exponents of periodic points in expanding circle maps.

Findings

Distribution of Lyapunov exponents approaches normal as period grows

Error in approximation decreases with increasing period

Provides a probabilistic understanding of periodic orbit stability

Abstract

For a smooth expanding circle map, we show that the empirical distribution of Lyapunov exponents of periodic points of any fixed period is close to normal, with an error that decreases as the period grows. This establishes a version of the Central Limit Theorem for such finite periodic orbits.