The growth speed of digits in infinite iterated function systems

Chun-Yun Cao¹ ; Bao-Wei Wang² ; Jun Wu²

¹ College of Science Huazhong Agricultural University 430070 Wuhan, P.R. China
² School of Mathematics and Statistics Huazhong University of Science and Technology 430074 Wuhan, P.R. China

Studia Mathematica, Tome 217 (2013) no. 2, pp. 139-158

Résumé

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 $\{f_n\}_{n\ge 1}$.

Détail

DOI : 10.4064/sm217-2-3

Keywords: geq infinite iterated function system satisfying set condition set varlambda its attractor varlambda except countably many points corresponds unique sequence integers called digit sequence lim rightarrow infty circ cdots circ investigate growth speed digits general infinite iterated function system precisely determine dimension set varlambda forall lim infty infty right infinite subset subset mathbb question posed hirst continued fractions generalize uczaks work dimension set varlambda text infinitely many mathbb see dimension sets above tightly connected convergence exponent contraction ratios sequence