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

Abstract

Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that $$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$ In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.

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).…”

“…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].…”