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## The growth speed of digits in infinite iterated function systems

### Tom 217 / 2013

Studia Mathematica 217 (2013), 139-158 MSC: Primary 11K55; Secondary 28A80. DOI: 10.4064/sm217-2-3

#### Streszczenie

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

#### Autorzy

• Chun-Yun CaoCollege of Science
Huazhong Agricultural University
430070 Wuhan, P.R. China
e-mail
• Bao-Wei WangSchool of Mathematics and Statistics
Huazhong University of Science
and Technology
430074 Wuhan, P.R. China
e-mail
• Jun WuSchool of Mathematics and Statistics
Huazhong University of Science
and Technology
430074 Wuhan, P.R. China
e-mail

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