Abstract. Strictly nonpointwise Markov operators are those which admit no nontrivial pointwise factors. The asymptotic set-system which is responsible for the pointwise factors is then weakly mixing. Examples of nontrivial behavior in this case are given.

INTRODUCTION

Many results proved for Markov operators, especially for their asymptotic behavior, are generalizations of analogous results obtained formerly for topological dynamical systems (see e.g. [Downarowicz–Iwanik, 1988], [Jamison–Sine, 1969], [Lotz, 1968]). A way of extending these results is by viewing the operator as a pointwise transformation on some compact space (the space of probability measures, the space of trajectories, etc.). Sometimes, however, this analogy fails to work. For example, the multiple recurrence theorem for n commuting operators has only been proved in the case of n = 2, since the passage to the space of trajectories used in the proof is not possible in higher dimensions [Downarowicz–Iwanik, 1988].

In order to better understand the nature of Markov operators in distinction from topological dynamical systems, this note is devoted to selecting and investigating operators whose behavior has as little in common with a pointwise transformation as possible. For most of the time, we will concentrate on the case of a minimal (irreducible) action of the operator. In Section 1, we introduce the definition of strictly nonpointwise Markov operators and derive a basic spectral property. Section 2 contains preliminaries for further deductions. In Section 3, the notion of the asymptotic set-system associated with a Markov operator is introduced, the carrier of the pointwise type behavior that still remains.