Algebraic dynamics and recursive inequalities

2024/02/20 by Junyi Xie, Xie, Junyi · 3 citations
Computer Science · #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2402.12678

openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We get three basic results in algebraic dynamics: (1). We give the first algorithm to compute the dynamical degrees to arbitrary precision. (2). We prove that for a family of dominant rational self-maps, the dynamical degrees are lower semi-continuous with respect to the Zariski topology. This implies a conjecture of Call and Silverman. (3). We prove that the set of periodic points of a cohomologically hyperbolic rational self-map is Zariski dense. Moreover, we show that, after a large iterate, every degree sequence grows almost at a uniform rate. This property is not satisfied for general submultiplicative sequences. Finally, we prove the Kawaguchi-Silverman conjecture for a class of self-maps of projective surfaces including all the birational ones. In fact, for every dominant rational self-map, we find a family of recursive inequalities of some dynamically meaningful cycles. Our proofs are based on these inequalities.

Cited by

• Cyclotomic integral points for affine dynamics
• Arithmetic Degrees are Cohomological Lyapunov Multipliers
• Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

Related

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