The topological entropy of endomorphisms of Lie groups

Abstract

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