This paper by Rufus Bowen studies topological entropy and its relationship with measure-theoretic entropy, focusing on algebraic examples such as Lie groups and homogeneous spaces. Topological entropy $ h_d(T) $ is defined for uniformly continuous maps on metric spaces, and it is shown to be equal to measure-theoretic entropy $ h(T) $ for Haar measure and affine maps on compact metrizable groups. For affine maps on Lie groups, the topological entropy is calculated in terms of eigenvalues, and a formula for the entropy of an affine map on a homogeneous space $ G/\Gamma $ is derived. The paper also discusses the relationship between topological entropy and measure-theoretic entropy for group endomorphisms and homogeneous spaces. Key results include the equality of topological and measure-theoretic entropy for certain maps, the computation of topological entropy for linear maps, and the extension of these results to homogeneous spaces. The paper also includes a proof of the product theorem for topological entropy and discusses the behavior of entropy under group extensions and flows. The results are applied to various examples, including toral automorphisms and endomorphisms of Lie groups.This paper by Rufus Bowen studies topological entropy and its relationship with measure-theoretic entropy, focusing on algebraic examples such as Lie groups and homogeneous spaces. Topological entropy $ h_d(T) $ is defined for uniformly continuous maps on metric spaces, and it is shown to be equal to measure-theoretic entropy $ h(T) $ for Haar measure and affine maps on compact metrizable groups. For affine maps on Lie groups, the topological entropy is calculated in terms of eigenvalues, and a formula for the entropy of an affine map on a homogeneous space $ G/\Gamma $ is derived. The paper also discusses the relationship between topological entropy and measure-theoretic entropy for group endomorphisms and homogeneous spaces. Key results include the equality of topological and measure-theoretic entropy for certain maps, the computation of topological entropy for linear maps, and the extension of these results to homogeneous spaces. The paper also includes a proof of the product theorem for topological entropy and discusses the behavior of entropy under group extensions and flows. The results are applied to various examples, including toral automorphisms and endomorphisms of Lie groups.