Lyapunov exponents of polynomials with respect to certain weighted Lyubich's measures

TL;DR

This paper computes Lyapunov exponents for hyperbolic polynomials on their Julia sets with respect to weighted Lyubich's measures, revealing properties of their coefficients and system behavior.

Contribution

It introduces a method to calculate Lyapunov exponents for specific polynomials and analyzes the coefficients' behavior, highlighting non-well-behavedness aspects.

Findings

Coefficients of Lyapunov exponents exhibit certain well-behavedness.

Quantification of non-well-behavedness in the system.

Analysis applies to monic, centered, hyperbolic polynomials.

Abstract

In this paper, we consider a monic, centred, hyperbolic polynomial of degree $d \ge 2$, restricted on its Julia set and compute its Lyapunov exponents with respect to certain weighted Lyubich's measures. In particular, we show a certain well-behavedness of some coefficients of the Lyapunov exponents, that quantifies the non-well-behavedness in a system.