# Disconnected maximal torus and the topological entropy of powers

The topological entropy is a measure of the chaoticity of a dynamical system generated by the iteration of a continuous transformation. When we iterate surjective endomorphisms of a Lie group $G$, its topological entropy coincides with that of its restriction to the maximal torus of the center of $G$. This implies that the entropy of endomorphisms of semi-simple compact Lie groups always vanishes. In this talk, we will determine the topological entropy of powers on Lie groups and show that it is always positive when the group is compact, even when it is semi-simple. We will focus on the role of the structure theory of Lie groups in proving this fact, especially on the role of the disconnected maximal torus, which we introduced and which fulfills the same role, in the case of disconnected compact groups, as the maximal torus in the connected case.