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Algebraic dynamics and recursive inequalities

Junyi Xie

TL;DR

This work advances algebraic dynamics by providing an explicit algorithm to compute all dynamical degrees $\lambda_i(f)$ to arbitrary precision using mixed degrees and intersection theory. It establishes the lower semi-continuity of dynamical degrees in families, with concrete consequences such as base-change stability and links to Call–Silverman type questions. A key technical development is the recursive-inequality framework for degree sequences, built on Siu-type inequalities, which yields sharp 2D lower bounds and underpins the density of periodic points for cohomologically hyperbolic maps. The authors then apply these tools to prove the Kawaguchi–Silverman conjecture for a broad class of self-maps on projective surfaces, including all birational maps, and they outline pathways to further algorithmic and arithmetic developments. Overall, the paper integrates birational modeling, recursive-degree analysis, and arithmetic dynamics to connect geometric growth, dynamical behavior, and number-theoretic consequences in a cohesive framework.

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.

Algebraic dynamics and recursive inequalities

TL;DR

This work advances algebraic dynamics by providing an explicit algorithm to compute all dynamical degrees to arbitrary precision using mixed degrees and intersection theory. It establishes the lower semi-continuity of dynamical degrees in families, with concrete consequences such as base-change stability and links to Call–Silverman type questions. A key technical development is the recursive-inequality framework for degree sequences, built on Siu-type inequalities, which yields sharp 2D lower bounds and underpins the density of periodic points for cohomologically hyperbolic maps. The authors then apply these tools to prove the Kawaguchi–Silverman conjecture for a broad class of self-maps on projective surfaces, including all birational maps, and they outline pathways to further algorithmic and arithmetic developments. Overall, the paper integrates birational modeling, recursive-degree analysis, and arithmetic dynamics to connect geometric growth, dynamical behavior, and number-theoretic consequences in a cohesive framework.

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.
Paper Structure (41 sections, 40 theorems, 170 equations)

This paper contains 41 sections, 40 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Dynamical degrees
  3. Cohomologically hyperbolic self-maps
  4. Algorithm to compute the dynamical degrees
  5. Our result
  6. Previous results
  7. The surface case
  8. Lower semi-continuity of dynamical degrees
  9. Strategy of the proof
  10. Periodic points
  11. Historical notes
  12. Strategy of the proof
  13. Kawaguchi-Silverman conjecture
  14. Further problems
  15. Birational models
  16. ...and 26 more sections

Key Result

Theorem 1.4

For any given number $l\in {\mathbb Z}_{>0},$ there is an explicit algorithm to output numbers $\widetilde{\lambda_i}, i=0,\dots, d$ such that $\lambda_i\in (\widetilde{\lambda_i}, \widetilde{\lambda_i}+\frac{1}{2^l})$, using finitely many mixed degrees.

Theorems & Definitions (58)

  • Example 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Example 1.10
  • Theorem 1.12
  • ...and 48 more