Measure-theoretic and topological entropy of operators on function spaces

Abstract

We study the entropy of actions on function spaces with the focus on doubly stochastic operators on probability spaces and Markov operators on compact spaces. Using an axiomatic approach to entropy we prove that there is basically only one reasonable measure-theoretic entropy notion on doubly stochastic operators. By ‘reasonable’ we mean extending the Kolmogorov–Sinai entropy on measure-preserving transformations and satisfying some obvious continuity conditions for $H_\mu$. In particular, this establishes equality on such operators between the entropy notion introduced by R. Alicki, J. Andries, M. Fannes and P. Tuyls (a version of which was also studied by I. I. Makarov), another notion of entropy introduced by E. Ghys, R. Langevin and P. Walczak, and our new definition introduced later in this paper. The key tool in proving this uniqueness is the discovery of a very general property of all doubly stochastic operators, which we call asymptotic lattice stability. Unlike the other explicit definitions of entropy mentioned above, ours satisfies many natural requirements already on the level of the function $H_\mu$, and we prove that the limit defining $h_\mu$ exists. The proof uses an integral representation of a stochastic operator obtained many years ago by A. Iwanik. In the topological part of the paper we introduce three natural definitions of topological entropy for Markov operators on C(X). Then we prove that all three are equal. Finally, we establish the partial variational principle: the topological entropy of a Markov operator majorizes the measure-theoretic entropy of this operator with respect to any of its invariant probability measures.