[Paper Review] Subshift of finite type and self-similar sets

Kan Jiang, Karma Dajani|arXiv (Cornell University)|Sep 16, 2015

Mathematical Dynamics and Fractals16 references3 citations

TL;DR

This paper establishes a correspondence between self-similar sets generated by iterated function systems (IFS) and subshifts of finite type under specific conditions—namely, when overlaps are exact and endpoints have periodic orbits. By identifying the self-similar set with a graph-directed self-similar set via a Markov partition, the authors compute the Hausdorff dimension of the set of points with unique codings and multiple codings using Mauldin-Williams theory, applying the method to $β$-expansions and survivor sets under the doubling map with asymmetrical holes.

ABSTRACT

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of $K$ as well as the set of elements in $K$ with unique codings using the machinery of Mauldin and Williams \cite{MW}. We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of $K$ with multiple codings. Secondly, in the setting of $β$-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik \cite[Theorem 1.8]{MK}. Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera \cite{Barrera}.

Motivation & Objective

• To establish conditions under which a self-similar set in $\mathbb{R}$ can be identified with a subshift of finite type.
• To compute the Hausdorff dimension of the set of points with unique codings and multiple codings in self-similar sets with overlaps.
• To extend the applicability of Mauldin-Williams theory to $\beta$-expansions and survivor sets under the doubling map with asymmetrical holes.
• To clarify the relationship between univoque sets and subshifts of finite type, showing that the set of unique codings is a subshift of finite type if and only if it is a sofic shift.

Proposed method

• Construct a Markov partition of the self-similar set $K$ by assuming exact overlaps and periodic orbits at endpoints of $f_i(K)$.
• Use the Markov partition to transform the full shift on $\{1,\dots,m\}^\mathbb{N}$ into a subshift of finite type via a transition matrix.
• Identify the self-similar set $K$ with a graph-directed self-similar set satisfying the open set condition due to the Markov structure.
• Apply the Mauldin-Williams theorem to compute the Hausdorff dimension of $K$ and of subsets such as the univoque set $U_F$.
• Use the doubling map with asymmetrical holes to model survivor sets, and define a Markov partition on intervals to establish isomorphism with a subshift of finite type.
• Define a bijection $\phi$ between the survivor set minus a countable set of preimages and a subshift of finite type, preserving dynamics via the shift map.

Experimental results

Research questions

• RQ1 Under what conditions can a self-similar set $K$ be identified with a subshift of finite type?
• RQ2 What is the Hausdorff dimension of the set of points in $K$ with exactly $k$ codings, and how does it vary with $k$?
• RQ3 Can the Hausdorff dimension of the univoque set be computed when the set of unique codings is not a subshift of finite type?
• RQ4 When is the set of unique codings a subshift of finite type, and how does this relate to sofic shifts?
• RQ5 What is a sufficient condition for the survivor set under the doubling map with asymmetrical holes to be isomorphic to a subshift of finite type?

Key findings

• For a specific IFS with four maps and $0 < \lambda < \frac{5 - \sqrt{21}}{2}$, the set $U_{2^k}$ has the same Hausdorff dimension as $U_1$, and $U_k = \emptyset$ for $k \neq 2^s$, $s \geq 1$, while $U_{\aleph_0} = \emptyset$.
• Any point in the self-similar set $K$ has either exactly $2^k$ codings for some $k \geq 0$, or uncountably many codings.
• The set of unique codings $\widetilde{U}_F$ is a subshift of finite type if and only if it is a sofic shift, a condition not previously noted by de Vries and Komornik.
• For the doubling map with asymmetrical holes $[a,b)$, the survivor set $J[a,b)$ has Hausdorff dimension $\frac{\log \alpha}{\log 2}$, where $\alpha$ is the spectral radius of the adjacency matrix $S'$.
• The survivor set $J[a,b)$ is measure-theoretically isomorphic to a subshift of finite type $\Sigma'$ via a Parry measure, after removing a countable set of preimages.
• In Example 5.9, with $a = \frac{1}{31}$, $b = \frac{2}{31}$, the spectral radius $\alpha$ of the adjacency matrix satisfies $\alpha^4 = \alpha^3 + \alpha^2 + \alpha + 1$, yielding $\dim_H(J(a,b)) = \frac{\log \alpha}{\log 2}$.