A topological conjugacy of invariant flows on some class of Lie groups

TL;DR

This paper establishes conditions under which invariant flows on certain Lie groups are topologically conjugate, showing that hyperbolic systems on these groups are conjugate, thus linking dynamical behavior to group structure.

Contribution

It provides a new criterion for topological conjugacy of invariant flows on Lie groups with associative or semisimple Lie algebras, focusing on hyperbolic systems.

Findings

Hyperbolic invariant flows are topologically conjugate on specified Lie groups.

Topological conjugacy depends on the hyperbolic nature of the systems.

Conditions for conjugacy are established for Lie groups with associative or semisimple Lie algebras.

Abstract

The aim of this paper is to give a condition to topological conjugacy of invariant flows in an Lie group $G$ which its Lie algebra $\mathfrak{g}$ is associative algebra or semisimple. In fact, we show that if two dynamical system on $G$ are hyperbolic, then they are topological conjugate.