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On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

Yohsuke Matsuzawa, Kaoru Sano

Abstract

We give a counterexample to the following conjecture: the set of isolated periodic points of an automorphism of degree at least two on an affine space is a set of bounded height. As a positive result, we prove that any cohomologically hyperbolic dominant rational self-map on a projective variety admits a non-empty Zariski open subset on which the set of periodic points is height bounded. Concerning preperiodic points, we give an example suggesting that the same statement may fail.

On the height boundedness of periodic and preperiodic points of dominant rational self-maps on projective varieties

Abstract

We give a counterexample to the following conjecture: the set of isolated periodic points of an automorphism of degree at least two on an affine space is a set of bounded height. As a positive result, we prove that any cohomologically hyperbolic dominant rational self-map on a projective variety admits a non-empty Zariski open subset on which the set of periodic points is height bounded. Concerning preperiodic points, we give an example suggesting that the same statement may fail.
Paper Structure (6 sections, 25 theorems, 126 equations)

This paper contains 6 sections, 25 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. An automorphism of A3 admitting periodic points with unbounded height
  3. height boundedness of periodic points
  4. Preperiodic points with unbounded height
  5. Dynamical degree
  6. Backward orbit of fixed points

Key Result

Theorem 1.1

Let Then, the following statements hold.

Theorems & Definitions (57)

  • Conjecture 1: silADS
  • Definition 1
  • Theorem 1.1
  • Definition 2
  • Definition 3
  • Theorem 1.2: Wang, Matsuzawa-Wang
  • Remark 1
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2
  • ...and 47 more