The growth speed of digits in infinite iterated function systems

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 thatWe investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the setfor 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 b n 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 .1. Introduction. We first recall the definition of an infinite iterated function system. For a thorough study and foundations of the theory of infinite iterated function systems, one is referred to the works of Hanus, Mauldin, and Urbański [HaMU], Mauldin and Urbański [MU1, MU2] or their monograph [MU3]. Here we follow the notation used in [JorR] by Jordan and Rams.Let {f n } n≥1 be a sequence of functions with(ii) Contraction property: there exists an integer m and a real number 0 < ρ < 1 such that for any (a 1 , . . . , a m ) ∈ N m and x ∈ [0, 1],(iii) Open set condition: for any i = j ∈ N, f i ((0, 1)) ∩ f j ((0, 1)) = ∅.

On Points With Positive Density of the Digit Sequence in Infinite Iterated Function Systems

“…(iv) ∞ n=1 f n ([0, 1]) = [0, 1] and, when i < j, f i (x) > f j (x). There is a natural projection Π : (1) for any a = {a n } n≥1 ∈ N N . Let Λ be the attractor of the iIFS { f n } n≥1 , that is to say,…”

Hausdorff dimension of certain sets arising in Engel continued fractions

“…It should be pointed out that regular continued fractions is a 2-decaying iIFS system in the context of Jordan and Rams [16]. See Cao et al [1], Liao and Rams [19], Zhang and Cao [26] for more general results from regular continued fractions to d-decaying iIFS systems (d > 1). In the present paper, we are interested in a variation of the regular continued fraction expansion, namely Engel continued fractions (ECFs).…”

“…If there exists some n ∈ N such that T k (x) = 0 for all k ≥ n, we say that the RCF expansion of x is finite and denote (1) by [a 1 (x), a 2 (x), · · · , a n (x)]. Otherwise, it is said to be infinite and denote (1) by [a 1 (x), a 2 (x), · · · , a n (x), · · · ]. It is known that a real number has an infinite RCF expansion if and only if it is irrational.…”

“…The problem on the Hausdorff dimension of E(φ) has been well improved in the context of iIFSs (see [1,16,26]). It should be pointed out that regular continued fractions is a 2-decaying iIFS system in the context of Jordan and Rams [16].…”

“…In the present paper, we are interested in a variation of the regular continued fraction expansion, namely Engel continued fractions (ECFs). We emphasize that ECFs can not be generated by some infinite iterated function system and hence that the general results on the Hausdorff dimension of E(φ) in the case of [1] and [16] can not be applied to ECFs, which is the main motivation of this paper. In 2002, Hartono et al [12] first introduced this new continued fraction algorithm with non-decreasing partial quotients.…”