Increasing digit subsystems of infinite iterated function systems

TL;DR

This paper studies subsets of infinite iterated function systems with increasing digit constraints, calculating their Hausdorff and packing dimensions, and generalizing previous results on continued fractions with increasing digits.

Contribution

It introduces a framework for analyzing the dimensions of digit-restricted subsets of infinite iterated function systems with polynomial contraction rates.

Findings

Calculated Hausdorff and packing dimensions of digit-restricted sets

Generalized Ramharter's result on continued fractions with increasing digits

Extended understanding of fractal dimensions in infinite IFS contexts

Abstract

We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$ if $i_n>\Phi(i_{n-1})$ for certain increasing functions $\Phi:\mathbb{N}\rightarrow\mathbb{N}$. We compute both the Hausdorff and packing dimensions of such sets. Our results generalize work of Ramharter which shows that the set of continued fractions with strictly increasing digits has Hausdorff dimension 1/2.