Lower bounds for non-Archimedean Lyapunov exponents

Abstract

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