Lower Bounds for non-Archimedean Lyapunov Exponents

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 $ϕ\in K(z)$ of degree $d\geq 2$ measures the exponential rate of growth along a typical orbit of $ϕ$. When $ϕ$ 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 $L(ϕ)$ for maps $ϕ$ defined over non-Archimedean fields $K$. The bound depends only on the degree $d$ and the Lipschitz constant of $ϕ$. For maps $ϕ$ whose Julia sets satisfy a certain boundedness condition, we are able to remove the dependence on the Lipschitz constant.