The growth speed of digits in infinite iterated function systems

Chun-Yun Cao, Bao-Wei Wang, and Jun Wu. "The growth speed of digits in infinite iterated function systems." Studia Mathematica 217.2 (2013): 139-158. <http://eudml.org/doc/285869>.

@article{Chun2013,
abstract = {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 $\{fₙ\}_\{n≥1\}$.},
author = {Chun-Yun Cao, Bao-Wei Wang, Jun Wu},
journal = {Studia Mathematica},
keywords = {infinite iterated function systems; digits; Hausdorff dimension},
language = {eng},
number = {2},
pages = {139-158},
title = {The growth speed of digits in infinite iterated function systems},
url = {http://eudml.org/doc/285869},
volume = {217},
year = {2013},
}

TY - JOUR
AU - Chun-Yun Cao
AU - Bao-Wei Wang
AU - Jun Wu
TI - The growth speed of digits in infinite iterated function systems
JO - Studia Mathematica
PY - 2013
VL - 217
IS - 2
SP - 139
EP - 158
AB - 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 ${fₙ}_{n≥1}$.
LA - eng
KW - infinite iterated function systems; digits; Hausdorff dimension
UR - http://eudml.org/doc/285869
ER -