Nonarchimedean Lyapunov exponents of polynomials

TL;DR

This paper investigates the Lyapunov exponents of polynomials over nonarchimedean fields, establishing bounds and nonnegativity conditions related to the polynomial's dynamics on the Berkovich Julia set.

Contribution

It provides new lower bounds for Lyapunov exponents of polynomials over nonarchimedean fields and explores conditions for their nonnegativity.

Findings

Lyapunov exponent has a lower bound depending only on degree and field.

Nonnegativity of Lyapunov exponents under tameness and no wandering Julia points.

Nonnegativity of critical value Lyapunov exponents when a unique Julia critical point exists.

Abstract

Let $K$ be an algebraically closed and complete nonarchimedean field with characteristic $0$ and let $f\in K[z]$ be a polynomial of degree $d\ge 2$. We study the Lyapunov exponent $L(f,\mu)$ of $f$ with respect to an $f$-invariant and ergodic Radon probability measure $\mu$ on the Berkovich Julia set of $f$ and the lower Lyapunov exponent $L_f^{-}(f(c))$ of $f$ at a critical value $f(c)$. Under an integrability assumption, we show $L(f,\mu)$ has a lower bound only depending on $d$ and $K$. In particular, if $f$ is tame and has no wandering nonclassical Julia points, then $L(f,\mu)$ is nonnegative; moreover, if in addition $f$ possesses a unique Julia critical point $c_0$, we show $L_f^{-}(f(c_0))$ is also nonnegative.