Algebraic Yuzvinski Formula

TL;DR

This paper establishes an algebraic analogue of the Yuzvinski Formula, linking the algebraic entropy of endomorphisms of finite-dimensional rational vector spaces to the Mahler measure of their characteristic polynomials.

Contribution

It proves the Algebraic Yuzvinski Formula, extending the classical topological entropy results to algebraic entropy for rational vector space endomorphisms.

Findings

Algebraic entropy equals Mahler measure of characteristic polynomial

Extension of Yuzvinski Formula to algebraic entropy

Discussion of applications and open problems

Abstract

In 1965 Adler, Konheim and McAndrew defined the topological entropy for continuous self-maps of compact spaces. Topological entropy is very well-understood for endomorphisms of compact Abelian groups. A fundamental result in this context is the so-called Yuzvinski Formula, showing that the value of the topological entropy of a full solenoidal automorphism coincides with the Mahler measure of its characteristic polynomial. In two papers of 1979 and 1981 Peters gave a different definition of entropy for automorphisms of locally compact Abelian groups. This notion has been appropriately modified forendomorphisms in two recent papers, where it is called algebraic entropy. The goal of this paper is to prove a perfect analog of the Yuzvinski Formula for the algebraic entropy, namely, the Algebraic Yuzvinski Formula, giving the value of the algebraic entropy of an endomorphism of a finite-dimensional rational vector space as the Mahler measure of its characteristic polynomial. Finally, several applications of the Algebraic Yuzvinski Formula and related open problems are discussed.