Substitution tilings with transcendental inflation factor

Substitution tilings with transcendental inflation factor, Discrete Analysis 2024:11, 24 pp.

Symbolic dynamics is the study of topological dynamical systems $(X,T)$, called subshifts, where $T\colon X\to X$ is the shift transformation acting on a shift-invariant subspace $X$ of the space $K^{\mathbb Z}$ of doubly infinite sequences in some set $K$, which is given the product topology. The set $K$ is called the alphabet of the shift.

Substitutions are special types of subshifts that exhibit a variety of phenomena and have been studied extensively when the alphabet $K$ is finite. The basic idea is that for each $a\in K$ there is a substitution rule of the form $a\mapsto a_1\cdots a_k$, and $X$ consists of the closure of the set of all sequences that are built up from elements of $K$ when we repeatedly apply these substitution rules.

Such symbolic substitutions can be made geometric, as so-called stone inflations of $\mathbb{R}$, in the following way. An interval tile is a finite subinterval of $\mathbb R$ (closed on the left and open on the right, say). Given a set of substitution rules, we assign to each letter $a$ a length $\ell(a)$. A tile inflation first inflates the length of each tile by a fixed inflation factor $\lambda>1$, resulting in a tile of length $\lambda \ell(a)$, and then decomposes the inflated tile in the obvious way into tiles of lengths $\ell(a_1),\dots,\ell(a_k)$, where $a\mapsto a_1\cdots a_k$ is the rule of the substitution. Repeating this process results in a tiling of $\mathbb{R}$, which is called a substitution tiling with inflation factor $\lambda$.

Substitutions on infinite alphabets and the corresponding tilings of $\mathbb{R}$, are natural objects of study in various settings, and their study has been steadily gaining attention. The goal of the current article is to better understand how the landscape of infinite alphabets differs from the more familiar world of finite alphabets. Previous works by N. Manibo, D. Rust, and J. Walton made a first foray into the spectral theory for infinite compact alphabets, and the current paper answers some of the questions raised in those works. In particular, the following question was raised: what inflation factors $\lambda$ can arise for a compact-alphabet substitution tiling of $\mathbb{R}$ with inflation symmetry (which means that if $x$ is an endpoint of one of the intervals, then so is $\lambda x$).

The inflation factor is the expansion factor of the substitution rule of the tiling. It is not too hard to show that for finite alphabets such expansion factors must be algebraic integers, but when the alphabet is infinite there are no obvious restrictions.

The main results of the paper establish criteria under which substitutions on infinite compact alphabets have certain statistical properties such as well-defined inflation factors, letter frequencies, and natural tile-length functions. Using the methodology they develop, the authors show that any real number greater than $2$ can be the expansion factor of a substitution tiling on the real line, which furthermore produces a uniformly discrete and relatively dense point set (a Delone set). Finally, an example of a substitution with an explicit transcendental inflation factor is constructed, making nice use of a classical transcendence result of Mahler.