FINITENESS OF COMMUTABLE MAPS OF BOUNDED DEGREE

In this paper, we study the relation between two dynamicalsystems (V,f) and (V,g) with f◦g= g◦f. As an application, we show thatan endomorphism (respectively a polynomial map with Zariski dense, ofbounded Preper(f)) has only finitely many endomorphisms (respectivelypolynomial maps) of bounded degree which are commutable with f. 1. IntroductionA dynamical system (V,f) consists of a set V and a self map f: V → V.If V is a subset of a projective space P n defined over a finitely generated fieldKover Q, then we have arithmetic height functions on V, which make a hugecontribution to the study of (V,f). In this paper, we show the following resultsby studying arithmetic relations between two dynamical systems (V,f) and(V,g) with f◦g= g◦f.Main Theorem (Theorems 3.3 and 4.2). (1) Let φ: P nC → P nC be an en-domorphism on P nC , of degree at least 2. Then there are only finitelymany endomorphisms of degree dwhich are commutable with φ:Com(φ,d) := {ψ∈ End(P nC ) | φ◦ψ= ψ◦φ, degψ= d}is a finite set.(2) Let f : A