Measure-theoretic and topological entropy of operators on function spaces

Abstract

Link to this article: http://journals.cambridge.org/abstract_S014338570400032XHow to cite this article: TOMASZ DOWNAROWICZ and BARTOSZ FREJ (2005). 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 µ . 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 µ , and we prove that the limit defining h µ 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.

Exploding Markov operators

“…This class of operators is a generalization of the following example described in [3]. It has positive entropy, yet it is strictly non-pointwise, meaning that the only pointwise factor of it is the trivial one (see section 3 for definitions).…”

“…k=i R(k) for every n, which ends the proof The definition of entropy of a doubly stochastic operator is not widely known, so I will devote next few lines for a short introduction to the subject-a detailed exposition may be found in [2] or [3] and an alternative approach in [7]. Similarly to the classical case of the Kolmogorov-Sinai invariant, the entropy of a doubly stochastic operator on L 1 (Y, ν) is defined in several steps.…”

“…Finally, h ν (F) is the supremum sup F h ν (P, F) over all collections under consideration. It was proved in [3] that any specification of the joining operation and the 'static' entropy H ν (F), which satisfies certain set of axioms, leads to a common value of the final notion h ν (P ). In addition, the conditional entropy of a collection F with respect to G is defined as…”

“…Moreover, the same symbol ∨ will be used for both the joining of partitions and the joining of collections of functions-in either case the meaning will be clear from the context. Below there is a list of some of the properties of entropy which can be found in [3]. (ii) For any finite collections F and G it holds that H ν (F ∨G) H ν (F)+H ν (G).…”

What does a generic Markov operator look like?

“…One definition was introduced, not quite distinctly, in [12]; this entropy is positive for generic polymorphisms of a finite space (bistochastic matrices). Another definition of the entropy of Markov operators, as well as further references, can be found in [14]. In ergodic theory, the generic value of the Kolmogorov entropy is zero.…”