Topological entropy of a Lie group automorphism

TL;DR

This paper investigates the topological entropy of automorphisms on certain connected Lie groups, revealing that entropy is contained within the toral component of the unstable subgroup in the radical.

Contribution

It establishes a relation between topological entropy and the structure of Lie groups with invariant Levi subgroups, extending understanding of entropy in this context.

Findings

Entropy is contained in the toral component of the unstable subgroup

Main results apply to finite semisimple center Lie groups with invariant Levi subgroups

Specialized results for particular Lie group structures

Abstract

Let {\phi} be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of {\phi} we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which admits a {\phi} invariant Levi subgroup. Then we prove that the topological entropy information of {\phi} is contained in the toral component of the unstable subgroup of {\phi} in the radical of G. We specialize the main result in a couple of interesting situations.