The growth speed of digits in infinite iterated function systems

• Autorzy: Chun-Yun Cao , Bao-Wei Wang , Jun Wu
• Języki publikacji: EN

Abstrakty

EN

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 ${fₙ}_{n≥1}$.

Kategorie tematyczne

• 28A80: Fractals
• 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2013
• Tom: 217
• Numer: 2
• Strony: 139-158

Daty

wydano

2013

Twórcy

autor

Chun-Yun Cao

• College of Science, Huazhong Agricultural University, 430070 Wuhan, P.R. China

autor

Bao-Wei Wang

• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P.R. China

autor

Jun Wu

• School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P.R. China

Identyfikatory

DOI

10.4064/sm217-2-3