Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

TL;DR

This paper investigates the growth rates of Birkhoff sums in infinite iterated function systems with polynomial decay, analyzing the Hausdorff dimensions of sets of points with similar sum growth behaviors.

Contribution

It provides new results on the Hausdorff dimensions of sets characterized by specific Birkhoff sum growth rates in Gauss-like systems with polynomial decay.

Findings

Hausdorff dimensions of sets with given Birkhoff sum growth rates are determined.

The study covers various unbounded potential functions.

Results extend understanding of dynamical complexity in systems with polynomial decay.

Abstract

The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.