A note on Zariski dense orbit conjecture

TL;DR

This paper discusses the Zariski dense orbit conjecture, presenting related results on birational automorphisms, periodic points, and specific cases for automorphisms of projective varieties with certain irregularity conditions.

Contribution

It provides new insights into ZDO by relating it to birational automorphisms and proving it for automorphisms of projective threefolds and varieties with high irregularity.

Findings

Bounded degree birational automorphisms relate to ZDO.

ZDO holds for automorphisms of projective threefolds.

ZDO is valid for varieties with irregularity q(X) ≥ dim X - 1.

Abstract

In this paper we first note a result of birational automorphisms with bounded degree of projective varieties related with the Zariski dense orbit conjecture (ZDO) and the Zariski density of periodic points. Next, we give a reduced result of ZDO for automorphisms of projective threefolds, and showed ZDO for automorphisms of projective varieties $X$ with the irregularity $q(X)\ge\dim X-1$.