Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps

2022/12/12 by Matsuzawa, Yohsuke, Wang, Long · 2 citations
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2212.05804

Abstract

A dominant rational self-map on a projective variety is called p-cohomologically hyperbolic if the p-th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over ℚ, we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if f is an 1-cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a ℚ-point with generic f-orbit is equal to the first dynamical degree of f; and (2) there are ℚ-points with generic f-orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental.

Cited by

• Rational self-maps of projective surfaces with a regular iterate
• Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

Related

• Arithmetic degree and its application to Zariski dense orbit conjecture
• Higher arithmetic degrees of dominant rational self-maps
• Periodic points and arithmetic degrees of certain rational self-maps
• Examples of dynamical degree equals arithmetic degree
• Arithmetic degrees and dynamical degrees of endomorphisms on surfaces