Minimally critical endomorphisms of P^N

TL;DR

This paper investigates the dynamics of certain endomorphisms on projective space, demonstrating that for high degrees, the critical height aligns with the moduli height, supporting a generalized Silverman conjecture.

Contribution

It establishes a relationship between critical height and moduli height for endomorphisms of projective space when the degree exceeds a specific threshold, extending Silverman's conjecture.

Findings

Critical height is comparable to moduli height for high-degree endomorphisms.

Supports a generalized form of Silverman's conjecture.

Provides conditions under which the relationship holds.

Abstract

We study the dynamics of the map endomorphism of N-dimensional projective space defined by f(X)=AX^d, where A is a matrix and d is at least 2. When d>N^2+N+1, we show that the critical height of such a morphism is comparable to its height in moduli space, confirming a case of a natural generalization of a conjecture of Silverman.