arXiv subjects

Natalie Priebe Frank

Publications and source records attributed to Natalie Priebe Frank.

15 recordsLinked to original sources

arXiv2025

(Don't) Mind the Gap

Infinite sequences are of tremendous theoretical and practical importance, and in the Information Age sequences of 0s and 1s are of particular interest. Over the past century, the field of symbolic dynamics has developed to study sequences with entries from a finite set. In this paper we will show you an interesting method for constructing sequences, and then we'll show you a fun variant of the method. The we'll discuss what theoretical computer scientists and mathematicians call the "complexity" of a sequence. We conclude by showing a technique you can use to compute the complexity of a sequence, illustrating it on our concrete examples.

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arXiv2024

Flow views and infinite interval exchange transformations for recognizable substitutions

A flow view is the graph of a measurable conjugacy $Φ$ between a substitution or S-adic subshift $(Σ,σ, μ)$ and an exchange of infinitely many intervals in $([0,1], F, m)$, where $m$ is Lebesgue measure. A natural refining sequence of partitions of $Σ$ is transferred to $([0,1],m)$ using a canonical addressing scheme, a fixed dual substitution, and a shift-invariant probability measure $μ$. On the flow view, $T \in Σ$ is shown horizontally at a height of $Φ(T)$ using colored unit intervals to represent the letters. The infinite interval exchange transformation $F$ is well approximated by exchanges of finitely many intervals, making numeric and graphic methods possible. We prove that in certain cases a choice of dual substitution guarantees that $F$ is self-similar. We discuss why the spectral type of $Φ\in L^2(Σ, μ),$ is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.

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arXiv2021

Spectral theory of spin substitutions

We introduce qubit substitutions in $\mathbb{Z}^m$, which have non-rectangular domains based on an endomorphism $Q$ of $\mathbb{Z}^m$ and a set $\mathcal{D}$ of coset representatives of $\mathbb{Z}^m/Q\mathbb{Z}^m$. We then focus on a specific family of qubit substitutions which we call spin substitutions, whose combinatorial definition requires a finite abelian group $G$ as its spin group. We investigate the spectral theory of the underlying subshift $(Σ,\mathbb{Z}^m)$. Under certain assumptions, we show that it is measure-theoretically isomorphic to a group extension of an $m$-dimensional odometer, which induces a complete decomposition of the function space $L^{2}(Σ,μ)$ . This enables one to use group characters in $\widehat{G}$ to derive substitutive factors and carry out a spectral analysis on specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous and singular continuous spectral measures, together with some bounds on their spectral multiplicity.

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arXiv2021

Introduction to hierarchical tiling dynamical systems

This paper is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional $Z^d$-systems, and multidimensional tiling systems. Aperiodically ordered structures are often hierarchical in nature, and there are a number of different yet related ways to define them. We will focus on what we are calling "supertile construction methods": symbolic substitution in one and many dimensions, S-adic sequences, self-similar and pseudo-self-similar tilings, and fusion rules. The techniques of dynamical analysis of these systems are discussed and a number of results are surveyed. We conclude with a discussion of the spectral theory of supertile systems from both the dynamical and diffraction perspectives.

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arXiv2020

Three variations on a theme by Fibonacci

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalisation structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

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arXiv2018

Fusion: a general framework for hierarchical tilings of $R^d$

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.

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arXiv2018

Fusion tilings with infinite local complexity

We propose a formalism for tilings with infinite local complexity (ILC), and especially fusion tilings with ILC. We allow an infinite variety of tile types but require that the space of possible tile types be compact. Examples include solenoids, pinwheel tilings, tilings with fault lines, and tilings with infinitely many tile sizes, shapes, or labels. Special attention is given to tilings where the infinite local complexity comes purely from geometry (shears) or comes purely from combinatorics (labels). We examine spectral properties of the invariant measures and define a new notion of complexity that applies to ILC tilings.

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arXiv2014

Fractal dual substitution tilings

Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing geometric and combinatorial, rather than topological, features of the original. Our method is easy to apply to particular substitution tilings, permits experimentation, and can be used to construct border-forcing substitution rules. For a large class of examples we show that the combinatorial dual tiling has a realization as a substitution tiling. Since the boundaries of our new tilings are fractal we are led to compute their fractal dimension. As an application of our techniques we show how to compute the Čech cohomology of a (not necessarily border-forcing) tiling using a graph iterated function system of a fractal tiling.

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arXiv2013

Tilings with infinite local complexity

This is a chapter surveying the current state of our understanding of tilings with infinite local complexity. It is intended to appear in the volume {\em Directions in Aperiodic Order}, D. Lenz, J. Kellendonk, and J. Savienen, eds.

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arXiv2013

Fusion: A general framework for hierarchical tilings

One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a different model for generating hierarchical tilings we call "fusion rules". Inflate-and-subdivide rules are a special case of fusion rules, but general fusion rules are more flexible and allow for defects, changes in geometry, and even constrained randomness. A condition that produces homogeneous structures and a method for computing frequency for fusion tiling spaces are discussed.

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arXiv2011

A fractal version of the pinwheel tiling

We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually locally derivable from the pinwheel tiling space. Interesting rotational properties, symmetries, and relative tile frequency are discussed for the tiling space associated with the fractal pinwheel tiling.

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arXiv2007

A dynamical system using the Voronoi tessellation

We introduce a dynamical system based on the vertices of Voronoi tessellations. This dynamical system acts on finite or discrete point sets in the plane, taking a point set to the vertex set of its Voronoi tessellation. We explore the behavior of this system for small point sets, then prove a general result quantifying the growth of the sizes of the point sets under iteration. We conclude by giving the most interesting open problems.

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arXiv2007

A primer on substitution tilings of the Euclidean plane

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to fairly represent the as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.

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arXiv2007

Topology of (some) tiling spaces without finite local complexity

A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Cech cohomology is related to properties of the fault lines.

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arXiv2005

Generalized $β$-expansions, substitution tilings, and local finiteness

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $β$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $β$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $β$-transformation.

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