The growth speed of digits in infinite iterated function systems

Chun-Yun Cao ; Bao-Wei Wang ; Jun Wu

Studia Mathematica, Tome 215 (2013), p. 139-158 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ${fₙ}_{n≥1}$.

Détail

Publié le : 2013-01-01

Zbl 1280.11043

EUDML-ID : urn:eudml:doc:285869