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## Studia Mathematica

2013 | 217 | 2 | 139-158
Tytuł artykułu

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

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Let ${fₙ}_{n≥1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${aₙ(x)}_{n≥1}$ of integers, called the digit sequence of x, such that
$x = lim_{n→∞} f_{a₁(x)}∘ ⋯ ∘ f_{aₙ(x)}(1)$.
We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set
${x ∈ Λ: aₙ(x) ∈ B (∀ n ≥ 1), lim_{n→∞} aₙ(x) = ∞}$
for any infinite subset B ⊂ ℕ, a question posed by Hirst for continued fractions. Also we generalize Łuczak's work on the dimension of the set
{x ∈ Λ: $aₙ(x) ≥ a^{bⁿ}$ for infinitely many n ∈ 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 ${fₙ}_{n≥1}$.
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Tom
Numer
Strony
139-158
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Daty
wydano
2013
Twórcy
autor
• College of Science, Huazhong Agricultural University, 430070 Wuhan, P.R. China
autor
• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P.R. China
autor
• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P.R. China
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