Approximation of non-archimedean Lyapunov exponents and applications over global fields

Abstract:

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-trivial and non-archimedean absolute value. We establish an approximation of the Lyapunov exponent of a rational map $f$ of $\mathbb {P}^1$ of degree $d>1$ defined over $K$ in terms of the multipliers of periodic points of $f$ having the formally exact period $n$, with an explicit error estimate in terms of $f,n$, and $d$. As an immediate consequence, we obtain an estimate on the blow-up of the Lyapunov exponent function near a pole in one-dimensional parameter families of rational maps over $K$.

Combined with an improvement of our former archimedean counterpart, this non-archimedean quantitative approximation of Lyapunov exponents allows us to establish

a quantification of Silverman’s and Ingram’s recent comparison between the critical height and any ample height on the dynamical moduli space $\mathcal {M}_d(\overline {\mathbb {Q}})$ except for the flexible Lattès locus,

an improvement of McMullen’s finiteness of the multiplier maps in two aspects: reduction to multipliers of cycles having a given formally exact period, and an explicit computation on the magnitude of the formally exact period of cycles, and

a characterization of non-affine isotrivial rational maps defined over a function field $\mathbb {C}(X)$ of a complex normal projective variety $X$ in terms of the growth of the degree of the multipliers of cycles.