The distribution of the largest digit for parabolic Iterated Function Systems of the interval

TL;DR

This paper studies the distribution of the largest digit in infinite parabolic IFSs on the interval, showing that the Hausdorff dimensions of level sets match that of the limit set, extending results from continued fractions.

Contribution

It establishes the Hausdorff dimension equality for level sets of largest digits in parabolic IFSs, generalizing previous theorems to non-uniformly expanding systems with infinitely many branches.

Findings

Hausdorff dimensions of level sets equal the limit set dimension

Results apply to backward and even integer continued fractions

Introduces a dimension theory for non-uniformly expanding Markov maps

Abstract

We investigate the distribution of the largest digit for a wide class of infinite parabolic Iterated Function Systems (IFSs) of the unit interval. Due to the recurrence to parabolic (neutral) fixed points, the dimension analysis of these systems become more delicate than that of uniformly contracting IFSs. We show that the Hausdorff dimensions of level sets associated with the largest digits are constantly equal to the Hausdorff dimension of the limit set of the IFS. This result is an analogue of Wu and Xu's theorem [Math. Proc. Camb. Phil. Soc. {\bf 146} (2009), 207--212] on the regular continued fraction. Examples of application of our result include the backward (aka minus, or negative) continued fractions, even integer continued fractions, and go beyond. Our main technical tool is a dimension theory for non-uniformly expanding Markov interval maps with infinitely many branches.