The topological entropy of endomorphisms of Lie groups

In this paper, we determine the topological entropy $h(\varphi )$ of a continuous endomorphism $\varphi$ 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 $h_d(\varphi )$ associated to a left invariant distance $d$ just provides an upper bound to $h(\varphi )$, which is characterized by the so called variational principle. We prove that $$h\left(\varphi \right)=h\left(\varphi {\mathrm{\mid }}_{T(G_{\varphi })}\right)$$ where $G_{\varphi }$ is the maximal connected subgroup of $G$ such that $\phi(G_\phi) = G_\phi$,and$T(G_\phi)$isthemaximaltorusinthecenterof$G_\phi$. 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.