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

Kostiantyn Drach, Zhi Fu, Vadim Kaloshin, Zhiqiang Li, Carlangelo Liverani

TL;DR

The paper proves a central limit theorem for the Lyapunov exponents of periodic points of fixed period in smooth expanding circle maps not smoothly conjugate to linear maps. It develops a transfer-operator framework to compute the average and variance of the periodic Lyapunov exponents and then uses spectral perturbation of a twisted operator to compare the resulting distribution to a standard normal, achieving a rate of convergence $\mathcal{O}(n^{-1/4})$. This builds a quantitative parallel with closed-geodesic length results on hyperbolic surfaces, highlighting a dictionary between dynamical length spectra and Lyapunov spectra. The results provide a rigorous statistical description of the length spectrum of periodic orbits in expanding circle maps and connect thermodynamic formalism with finite-period orbit statistics.

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.

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

TL;DR

The paper proves a central limit theorem for the Lyapunov exponents of periodic points of fixed period in smooth expanding circle maps not smoothly conjugate to linear maps. It develops a transfer-operator framework to compute the average and variance of the periodic Lyapunov exponents and then uses spectral perturbation of a twisted operator to compare the resulting distribution to a standard normal, achieving a rate of convergence . This builds a quantitative parallel with closed-geodesic length results on hyperbolic surfaces, highlighting a dictionary between dynamical length spectra and Lyapunov spectra. The results provide a rigorous statistical description of the length spectrum of periodic orbits in expanding circle maps and connect thermodynamic formalism with finite-period orbit statistics.

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.

Paper Structure

This paper contains 5 sections, 4 theorems, 49 equations, 1 figure.

Table of Contents

  1. Main result
  2. Preliminaries
  3. The average
  4. The variance
  5. The distribution and proof of the Main Theorem

Key Result

Lemma 3.1

For every $z\in \mathbb{T}$ and every integer $n \geqslant1$,

Figures (1)

  • Figure 1: The empirical distributions of the Lyapunov exponents for periodic orbits of period $30$, using $100$ equal-width bins, where the map is: (A) $x \mapsto 2x+0.01 (\sin(2\pi x) + \cos(2\pi x) - 1) \mod 1$; (B) the Blaschke product $z \mapsto z (z-0.1)/(1-0.1 z)$ restricted to the unit circle; in this case, the corresponding expanding circle map preserves the Lebesgue measure on the circle.

Theorems & Definitions (6)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Corollary 3.3: Centered expectation estimate
  • Lemma 4.1: Centered variance estimate
  • proof