Vol. 96, No. 2, 1981

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Entropy of automorphisms on L.C.A. groups

Justin Peters

Vol. 96 (1981), No. 2, 475–488

In this paper we will consider entropy of automorphisms on locally compact abelian groups. Bowen’s definition of entropy of a uniformly continuous mapping applies in particular to topological automorphisms of l.c.a. groups. If hB(α,G) denotes the Bowen entropy of α Aut(G), we investigate the appropriate dual notion h(α,Ĝ) of the adjoint automorphism α on the dual group Ĝ, and show hB(α,G) = h(α,Ĝ). We define the total entropy h(α,G) of α on G to be the sum hB(α,G) + h(α,G) and show that with this definition, h(α,G) coincides with Kolmogorov-Sinai entropy if G is compact and furthermore the invariance properties present in the compact case are retained for an arbitrary l.c.a. group G. We also obtain the addition theorem for entropy and a formula for the entropy on projective limits. In conclusion we mention some questions which arise.

Mathematical Subject Classification 2000
Primary: 54H20
Secondary: 22D40, 28D20, 54C70, 58F11
Received: 1 June 1979
Revised: 6 June 1980
Published: 1 October 1981
Justin Peters
Department of Mathematics
Iowa State University
Ames IA 50011
United States