Lower Bounds for non-Archimedean Lyapunov Exponents

TL;DR

This paper establishes lower bounds for Lyapunov exponents of rational maps over non-Archimedean fields, extending classical bounds and identifying conditions under which the bounds depend solely on degree.

Contribution

It provides new lower bounds for non-Archimedean Lyapunov exponents that depend only on degree and Lipschitz constant, with special cases removing the Lipschitz dependence.

Findings

Lower bounds depend on degree and Lipschitz constant.

For certain Julia sets, bounds depend only on degree.

Classical bound of 0.5 log d is extended to non-Archimedean settings.

Abstract

Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\textbf{P}^1$ denote the Berkovich projective line over $K$. The Lyapunov exponent for a rational map $\phi\in K(z)$ of degree $d\geq 2$ measures the exponential rate of growth along a typical orbit of $\phi$. When $\phi$ is defined over $\mathbb{C}$, the Lyapunov exponent is bounded below by $\frac{1}{2}\log d$. In this article, we give a lower bound for $L(\phi)$ for maps $\phi$ defined over non-Archimedean fields $K$. The bound depends only on the degree $d$ and the Lipschitz constant of $\phi$. For maps $\phi$ whose Julia sets satisfy a certain boundedness condition, we are able to remove the dependence on the Lipschitz constant.