Lower bounds for non-Archimedean Lyapunov exponents

Abstract:

Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\mathrm {\mathbf {P}}^1$ 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.