Abstract. Topological entropy há(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare hd(T) with measure theoretic entropy h(T); in particular h(T) = hd(T) for Haar measure and affine maps Ton compact metrizable groups. A particular case of this yields the wellknown formula for h(T) when T is a toral automorphism.
The topological entropy of a continuous map on a compact space was defined by Adler, Konheim and McAndrew [l]. In the present paper we will define entropy for subsets of compact spaces in a way which resembles Hausdorff dimension.This will be used to genetalize known results about the Hausdorff dimension of the quasiregular points of certain measures and to define a notion of conjugacy that is a cross between the topological and measure theoretic ones.In [5] we gave a definition of entropy for uniformly continuous maps on metric spaces. That definition was motivated by different examples (linear maps on R"and calculating entropy on T") and it sometimes differs from the definition given here.We wish to thank Karl Sigmund who pointed us in the direction this paper takes and Ben Weiss who helped us formulate §4.
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