Arithmetic Degrees are Cohomological Lyapunov Multipliers

TL;DR

This paper establishes a link between arithmetic degrees and cohomological Lyapunov multipliers for endomorphisms of projective varieties, providing insights that could impact the understanding of the dynamical Mordell--Lang conjecture.

Contribution

It proves that the arithmetic degree of a point with Zariski dense orbit equals a cohomological Lyapunov multiplier, a novel connection in algebraic dynamics.

Findings

Arithmetic degree equals a cohomological Lyapunov multiplier for dense orbits

Application to the dynamical Mordell--Lang conjecture

Provides a new tool for studying algebraic dynamical systems

Abstract

For endomorphisms of projective varieties, we prove that the arithmetic degree of a point with Zariski dense orbit must be a cohomological Lyapunov multiplier of the dynamical system. We will apply our result to deduce a corollary towards the dynamical Mordell--Lang conjecture.