Rigidité, expansion et entropie en dynamique non-archimédienne (Rigidity, expansion and entropy in non-Archimedean dynamics)
Charles Favre, Juan Rivera-Letelier
TL;DR
The paper proves a non-archimedean rigidity phenomenon: if the equilibrium measure ${\rho_R}$ of a rational map on the Berkovich line charges a segment of the hyperbolic tree ${\mathbb{H}_K}$, then the map is affine Bernoulli, mirroring Zdunik-type rigidity in the complex setting. It builds a finite-length affine model via a quotient ${\mathcal{T}}$ of ${\mathsf{P}^{1}_K}$, showing the induced dynamics is piecewise affine with a segment structure, and deduces that in this regime the topological entropy is the logarithm of an integer. Beyond rigidity, the work analyzes Lyapunov exponents through a geometric sub-cobord framework, linking positive exponents to full support on ${\mathbb{H}_K}$ and relating zero-exponent cases to good/bad reduction and affine Bernoulli behavior. The results extend to applications for moderated rational maps and meromorphic complex families, yielding entropy discretization, equidistribution properties for repelling cycles, and structural insights into ramification wildness. Overall, the paper fuses Berkovich potential theory with dynamical systems to produce sharp rigidity statements and entropy characterizations in non-archimedean dynamics.
Abstract
We prove a rigidity property in non-Archimedean dynamics, reminiscent of Zdunik theorem in complex dynamics: every rational map whose equilibrium measure charges an interval in the Berkovich projective line is affine Bernoulli. Our proof is inspired by the construction of the affine model of multimodal maps by Parry and by Milnor and Thurston. This rigidity result allows us to show that the topological entropy of any tame rational map is the logarithm of an integer. To that end we analyze the properties of the multiplicative (sub)coboundary given by the spherical derivative and we establish a link between the sign of the Lyapunov exponent and the locus of wild ramification.