The growth speed of digits in infinite iterated function systems
Tom 217 / 2013
Streszczenie
Let 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}.
