Entropy of Endomorphisms of Lie Groups

TL;DR

This paper investigates the entropy of surjective endomorphisms of Lie groups, showing it depends on the toral component of the center, with specific results for semi-simple and compact groups.

Contribution

It establishes a formula for entropy of endomorphisms of Lie groups based on their central toral components, extending understanding of entropy in these structures.

Findings

Entropy equals the entropy of restriction to the toral center component for nilpotent and reductive groups.

Entropy vanishes for surjective endomorphisms of semi-simple Lie groups.

Entropy formula simplifies to that of a torus for compact groups.

Abstract

We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus.