# The dependence of Lyapunov exponents of polynomials on its coefficients

## TL;DR

This paper investigates how Lyapunov exponents of quadratic and cubic polynomials vary with their parameters, using thermodynamic formalism to analyze their dependence on complex coefficients over Julia sets.

## Contribution

It introduces a method to compute Lyapunov exponents for polynomial families using thermodynamic formalism, revealing parameter dependence on these exponents.

## Key findings

- Lyapunov exponents vary smoothly with polynomial parameters.
- Dependence patterns differ between quadratic and cubic families.
- Results are obtained for various Bernoulli measures.

## Abstract

In this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely $P_{c}(z) = z^{2} + c$ with $|c| < 1$ and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely $P_{(a_{1},a_{0})}(z) = z^{3} + a_{1}z + a_{0}$ with $|a_{i}| < 1$, restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtain results pertaining to the dependence of the behaviour of these exponents on the parameters describing the polynomial map. We achieve this using the theory of thermodynamic formalism, the pressure function in particular.