A remark on big Birkhoff sums in d-decaying Gauss like iterated function systems

Abstract

Recently, Liao and Rams [10] considered certain infinite iterated function systems that are natural generalizations of Gauss map. They studied the increasing rate of Birkhoff sums in such systems with polynomial decay of the derivative and calculated the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate for different unbounded potential functions. In this remark, we investigate the critical cases for the growth rate function which are not discussed in [10].