A Chung--Fuchs type theorem for skew-product dynamical systems

Abstract

We prove a Chung--Fuchs type theorem for skew-product dynamical systems with invertible base. Suppose that the Birkhoff averages of a measurable function converge to zero almost surely and that its successive iterates are independent on almost every fibre, with a nonconstant conditional law for the function itself. Then the Birkhoff sums are unbounded above with positive probability and unbounded below with positive probability. If the skew product is ergodic, they oscillate almost surely. As an application, we prove that Mandelbrot cascades acting on the fibre measures of the natural extension of an ergodic symbolic measure are degenerate in the critical case, provided that the weights are not of ${\mathcal{Q}}_{\alpha }$-type.

arXiv comment8 pages