Increasing digit subsystems of infinite iterated function systems

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.