[Paper Review] Multiple codings for self-similar sets with overlaps

Karma Dajani, Kan Jiang+3 more|arXiv (Cornell University)|Mar 30, 2016

Mathematical Dynamics and Fractals33 references16 citations

TL;DR

This paper investigates the structure of self-similar sets with complete overlaps on the real line, focusing on the number of codings (expansions) for each point. It establishes that the set of points with exactly one coding is closed if and only if there are no points with countably infinite codings, provides explicit formulas for the Hausdorff dimension of sets with k codings, proves that the Hausdorff measure of such sets is infinite for k ≥ 2, and computes the local dimension of self-similar measures at points with k or infinitely many codings.

ABSTRACT

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $Φ=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n o\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(Φ)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(Φ)$ is closed if and only if $\mathcal U_{\aleph_0}(Φ)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(Φ)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(Φ)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(Φ)$ and ${U_{\aleph_0}(Φ)}$.

Motivation & Objective

• To analyze the structure of self-similar sets with complete overlaps, particularly the distribution of points with multiple codings.
• To characterize the topological properties of sets with exactly k codings, especially the closure of the unique coding set.
• To compute the Hausdorff dimension and measure of the sets of points with exactly k codings.
• To determine the local dimension of self-similar measures at points with k or countably infinite codings.
• To extend results on non-integer base expansions and Bernoulli convolutions to a broader class of self-similar systems with overlaps.

Proposed method

• The authors define a class E of self-similar iterated function systems (IFS) with complete overlaps, where overlapping occurs between adjacent maps.
• They classify points in the attractor E by the number of distinct codings: U_k(Φ) denotes the set of points with exactly k different codings.
• Using geometric and combinatorial techniques, they derive explicit formulas for the Hausdorff dimension of U_k(Φ) based on the contraction ratios and overlaps.
• They analyze the local dimension of self-similar measures by studying the growth rate of measure in small balls around points in U_k(Φ) and U_ℵ₀(Φ), using recursive inequalities and asymptotic analysis.
• The proof involves constructing nested intervals and estimating measure growth via sequences of maps, leading to limits involving logarithms of probabilities and contraction ratios.
• They establish a duality between the closure of U_1(Φ) and the emptiness of U_ℵ₀(Φ), using topological and metric arguments.

Experimental results

Research questions

• RQ1 Under what conditions is the set of points with a unique coding closed in a self-similar set with complete overlaps?
• RQ2 What is the Hausdorff dimension of the set of points with exactly k codings in such systems?
• RQ3 Why is the Hausdorff measure of the set of points with k ≥ 2 codings always infinite?
• RQ4 How does the local dimension of a self-similar measure vary at points with k codings or countably infinitely many codings?
• RQ5 Can the local dimension be explicitly computed in terms of the probabilities and contraction ratios of the IFS?

Key findings

• The set U_1(Φ) is closed if and only if U_ℵ₀(Φ) is empty, establishing a topological dichotomy for the system.
• The Hausdorff dimension of U_k(Φ) is given by an explicit formula depending on the contraction ratios and overlap structure, with the dimension being strictly less than that of the full attractor E for k ≥ 2.
• For all k ≥ 2, the Hausdorff measure of U_k(Φ) is infinite, indicating that these sets are large in a metric sense despite being small in topological dimension.
• The local dimension of the self-similar measure μ_p at points in U_k(Φ) is given by a min-max formula involving the probabilities and contraction ratios of the maps involved in the coding.
• At points in U_ℵ₀(Φ), the local dimension is determined by the infimum of a set of candidate values derived from infinite coding sequences, reflecting the complexity of the expansion structure.
• The local dimension is constant on each U_k(Φ) and U_ℵ₀(Φ), and can be computed as the limit of log μ_p(B(x,r)) / log r as r → 0.