The topological entropy of endomorphisms of Lie groups

TL;DR

This paper determines the topological entropy of continuous endomorphisms of Lie groups, reducing the problem to the entropy of a torus, and clarifies the relationship between entropy and Li-Yorke pairs.

Contribution

It provides a formula for the topological entropy of endomorphisms of Lie groups, linking it to the entropy on a maximal torus, and clarifies the entropy's relation to group structure.

Findings

Topological entropy equals that of the restriction to the maximal torus in the center.

Entropy computation reduces to the classical formula for tori.

Null entropy relates to the absence of Li-Yorke pairs.

Abstract

In this paper, we determine the topological entropy $h(\phi)$ of a continuous endomorphism $\phi$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(\phi)$ associated to a left invariant distance $d$ just provides an upper bound to $h(\phi)$, which is characterized by the so called variational principle. We prove that \[ h\left(\phi\right) = h\left(\phi|_{T(G_\phi)}\right) \] where $G_\phi$ is the maximal connected subgroup of $G$ such that $\phi(G_\phi) = G_\phi$, and $T(G_\phi)$ is the maximal torus in the center of $G_\phi$. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.