The growth speed of digits in infinite iterated function systems
Volume 217 / 2013
Abstract
Let $\{f_n\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ satisfying the open set condition with the open set $(0,1)$ and let $\varLambda $ be its attractor. Then to any $x\in \varLambda $ (except at most countably many points) corresponds a unique sequence $\{a_n(x)\}_{n\ge 1}$ of integers, called the digit sequence of $x$, such that $$ x=\lim_{n\rightarrow \infty }f_{a_1(x)}\circ \cdots \circ f_{a_n(x)}(1). $$ We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set $$ \left \{x\in \varLambda : a_n(x)\in B \ (\forall n\ge 1), \lim_{n\to \infty }a_n(x)=\infty \right \} $$ for any infinite subset $B\subset \mathbb N$, a question posed by Hirst for continued fractions. Also we generalize Łuczak's work on the dimension of the set $$ \{x\in \varLambda : a_n(x)\ge a^{b^n} \ \text {for infinitely many}\ n\in \mathbb N\} $$ with $a,b>1$. We will see that the dimension of the sets above is tightly connected with the convergence exponent of the contraction ratios of the sequence $\{f_n\}_{n\ge 1}$.
