Finiteness of commutable maps of bounded degree

TL;DR

This paper investigates the finiteness of maps commuting with a given dynamical system, establishing that only finitely many such maps of bounded degree exist under certain conditions.

Contribution

It proves the finiteness of commutable endomorphisms and polynomial maps of bounded degree for a given dynamical system with specific properties.

Findings

Finitely many endomorphisms commute with a given map under certain conditions.

Finitely many polynomial maps of bounded degree commute with a given polynomial map.

Results apply to maps with Zariski dense or bounded preperiodic points.

Abstract

In this paper, we study the relation between two dynamical systems (V,f) and (V,g) with f. g = g . f. As an application, we show that an endomorphism (respectively a polynomial map with Zariski dense, of bounded Pre(f) has only finitely many endomorphisms (respectively polynomial maps) of bounded degree which are commutable with f.