The Lyapunov exponent of holomorphic maps

TL;DR

This paper investigates the relationship between Lyapunov exponents and attracting cycles in complex dynamical systems, establishing conditions under which the exponents are negative or zero, and analyzing typical behavior in Julia sets.

Contribution

It proves a characterization of negative Lyapunov exponents at critical values for polynomial and exponential maps, and analyzes Lyapunov exponents for points in Julia sets of unicritical polynomials.

Findings

Negative Lyapunov exponent at critical value iff the map has an attracting cycle.

Almost every point in Julia sets of unicritical polynomials with positive area has zero Lyapunov exponent.

Points with positive upper Lyapunov exponent are not Lebesgue density points in Julia sets.

Abstract

For any polynomial map with a single critical point, we prove that its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.