On uniform convergence in ergodic theorems for a class of skew product transformations

Abstract

Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space. The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function F on the product of the two probability spaces along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity andintegrability conditions for F, we prove uniform convergence in the base and L^p(P)-convergence in the fibre. Under an equicontinuity assumption on F we further show P-almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns defined by a direction.