Nick Ramsey
We bound the change in entropy incurred by an irreducible subshift of finite type upon perturbing it by forbidding a pair of admissible words. Lind has proven such bounds in the one-word case, and we adapt his methods. In particular, we introduce multi-word correlation polynomials and study their size, as well as that of their determinant in the two-word case.
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Given an SFT $Σ$ and a finite set $S$ of finite words, let $Σ\langle S\rangle$ denote the subshift of $Σ$ that avoids $S$. We establish a general criterion under which we can bound the entropy perturbation $h(Σ)-h(Σ\langle S\rangle)$ from above. As an application, we prove that this entropy difference tends to zero with a sequence of such sets $S_1, S_2, \dots$ under various assumptions on the $S_i$.
We introduce a class of coded systems that we construct from sofic systems and Dyck shifts and we study a class of subshifts that we obtain by excluding words of length two from Dyck shifts. We derive expressions for zeta functions and topological entropy. We derive an expression for the zeta function of certain subshifts that we obtain by excluding words from Dyck shifts and of certain subshifts that we obtain by excluding words from the subshifts that are constructed from full shifts and Dyck shifts.
On finite spacer rank for words and subshifts
本文扩展了Gao和Hill的秩一符号移位概念，提出了字词与子移位的“间隔符秩”，构造了不同秩的无穷字词，给出了代换不动点至多间隔符秩二的条件，并证明了有限间隔符秩子移位拓扑熵为零及相关性质。
Measure-Theoretically Mixing Subshifts of Minimal Word Complexity
本文解决了测度论动力复杂度与符号复杂度的长期未解问题，确定了强混合出现的最小词复杂度：超线性复杂度可支持所有阶强混合，非超线性复杂度则导致部分刚性。
Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts
本文研究一阶子位移的词复杂度，构造了容许弱混合测度且复杂度渐近接近1.5q的子位移，并证明非里程表一阶变换对应的子位移满足limsup p(q)-1.5q=∞，且该结果最优。
On word complexity and topological entropy of random substitution subshifts
本文研究随机替换子位移的词复杂性与拓扑熵，不假设相容性，证明原始随机替换零熵等价于确定性替换，并给出常长度情形熵的计算方法及复杂度函数的分类。
We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.
A note on the perturbations of subshifts
本文研究了不同类别子位移在禁止包含特定词的序列时的扰动性质，证明了sofic shift的扰动仍为sofic shift，给出了synchronized system保持同步的充分条件，并显式计算了$\mathcal{S}$-gap shift和有限型子位移在扰动下的熵降。
Any coded subshift X defined by a set C of code words contains a subshift, which we call L, consisting of limits of single code words. We show that when C satisfies a unique decomposition property, the topological entropy h(X) of X is determined completely by h(L) and the number of code words of each length. More specifically, we show that h(X) = h(L) exactly when a certain infinite series is less than or equal to 1, and when that series is greater than 1, we give a formula for h(X_C). In the latter case, an immediate corollary (using results of Climenhaga and Thompson) is that X has a unique …
On subshifts with slow forbidden word growth
本文研究禁词数量缓慢增长的子位移，证明其具有有界超可乘性和唯一最大熵测度，并推广到α-β位移等已知类。
On minimal subshifts of linear word complexity with slope less than 3/2
本文证明词复杂度上渐近斜率小于 3/2 的无限极小子移位测度同构于最大等连续因子，并由此推出 Sarnak 猜想，同时用 S-adic 分解刻画其结构。
This article studies the complexity of the word problem in groups of automorphisms of subshifts. We show in particular that for any Turing degree, there exists a subshift whose automorphism group contains a subgroup whose word problem has exactly this degree.
A minimal subshift $(X,T)$ is linearly recurrent if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word $u$ the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a linearly recurrent subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts.
We construct a family of ergodic measures on random substitution subshifts (RS-subshifts) associated to a primitive random substitution. In particular, the word frequencies of every finite legal word exist for almost every element of the random substitution subshift with respect to these measures. As an application, we show that for a certain class of random substitutions the measures of maximal entropy are frequency measures.
We study the class of word-building games, where two players pick letters from a finite alphabet to construct a finite or infinite word. The outcome is determined by whether the resulting word lies in a prescribed set (a win for player $A$) or not (a win for player $B$). We focus on symbolic dynamical games, where the target set is a subshift. We investigate the relation between the target subshift and the set of turn orders for which $A$ has a winning strategy.
This paper examines the relationship between the escape rate and the minimal period of the hole. We consider a subshift of finite type on $q$ symbols with a union of $t$ cylinders based at words of identical length $p$ as the hole. The escape rate into the hole relates to the asymptotic behavior of the number of words of fixed length that do not contain the fixed set of (forbidden) words at which the cylinders are based. We explore the relationship between the escape rate and $r(z)$, a rational function of the correlations between the forbidden words. In particular, we prove that there exists …
In this work we consider several aspects of the thermodynamic formalism in a randomized setting. Let $X$ be a non-trivial mixing shift of finite type, and let $f : X \to \mathbb{R}$ be a Hölder continuous potential with associated Gibbs measure $μ$. Further, fix a parameter $α\in (0,1)$. For each $n \geq 1$, let $\mathcal{F}_n$ be a random subset of words of length $n$, where each word of length $n$ that appears in $X$ is included in $\mathcal{F}_n$ with probability $1-α$ (and excluded with probability $α$), independently of all other words. Then let $Y_n = Y(\mathcal{F}_n)$ be the random subs…
It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $\mathbb{Z}$ [DDPM15, CK15]. In this article we extend this result to a broader class proving that the automorphism group of a minimal S-adic subshift of finite alphabet rank is virtually $\mathbb{Z}$. The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.
We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree 0 or is a union of c…
Low Complexity Subshifts have Discrete Spectrum
本文证明了当线性字复杂度的上极限 $C < \frac{4}{3}$ 时，子移位具有离散谱，并给出了 $C = \frac{3}{2}$ 时存在弱混合测度的例子，将此类子移位下确界的范围缩小至 $[\frac{4}{3}, \frac{3}{2}]$。
A subshift of finite type over finitely many symbols can be described as a collection of all infinite walks on a digraph with at most a single edge from a vertex to another. The associated finite set $\F$ of forbidden words is a constraint which determines the language of the shift entirely. In this paper, in order to describe infinite walks on a multigraph, we introduce the notion of multiplicity of a word (finite walk) and define repeated words as those having multiplicity at least $2$. In general, for given collections $\F$ of forbidden words and $\R$ of repeated words with pre-assigned mul…
On the complexity of subshifts and infinite words
本文完整刻画了子移位复杂度函数的渐近等价类，证明非递减、次可乘且至少线性增长是递归子移位复杂度函数的充要条件，并给出显式算法构造。
We construct a binary minimal subshift whose words of length n form a connected subset of the Hamming graph for each n.
We study subshift that arise by excluding words of length two from Dyck shifts. The words that are to be excluded are taken from a finite set that is not literal-uniform.
In this work, we study the problem of finding the asymptotic growth rate of the number of of $d$-dimensional arrays with side length $n$ over a given alphabet which avoid a list of one-dimensional "forbidden" words along all cardinal directions, as both $n$ and $d$ tend to infinity. Louidor, Marcus, and the second author called this quantity the "limiting entropy"; it is the limit of a sequence of topological entropies of a sequence of isotropic $\mathbb{Z}^d$ subshifts with the dimension $d$ tending to infinity. We find an expression for this limiting entropy which involves only one-dimension…
An infinite word x is said to be quasiperiodic if there exists a finite word q such that x is covered by occurrences of q (such a q is called a quasiperiod of x). Using the notion of derivation, we show that this definition is not sufficient to imply any symmetry in an infinite word. Therefore we introduce multi-scale quasiperiodic words, i.e. quasiperiodic words that admit an infinite number of quasiperiods. Such words are uniformly recurrent, this allows us to study the subshift they generate. We prove that multi-scale quasiperiodic subshifts are uniquely ergodic and have zero topological en…
In this paper we fix a set Λ^* of positive elements of the free group F_n (e.g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S:F_n --> L(H) which we prove to be a partial representation of F_n on H under certain conditions studied by Matsumoto.
In this paper, we describe the relationship between the Perron root and eigenvectors of an irreducible subshift of finite type with the correlation between the forbidden words in the subshift. In particular, we derive an expression for the Perron eigenvectors of the associated adjacency matrix. As an application, we obtain the Perron eigenvectors for irreducible $(0,1)$ matrices which are adjacency matrices for directed graphs. Moreover, we derive an alternate definition of the Parry measure in ergodic theory on an irreducible subshift of finite type.
Minimal zero entropy subshifts can be unrestricted along any sparse set
该研究简化证明了零Banach密度稀疏集上极小零熵子移位可呈现任意行为，给出更通用的多项式Sarnak猜想反例，说明仅极小性和零熵假设无法推出相关结论。
Freezing phase transition for the Thue-Morse subshift
本文研究了两个符号全移位上与 Thue-Morse 替换相关的势函数，证明了当参数足够大时存在压强为零的冻结相变，并修正了前人证明中的重大漏洞。
On subshifts with low maximal pattern complexity
本文通过最大等连续因子的拓扑性质，将递归模式 Sturmian 序列分类为无理圆周旋转的两区间编码或近简单 Toeplitz 子移位，并给出非超线性最大模式复杂度序列的一般结构结果。
Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $ω$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $α$. Let $X_ω$ be the (random) SFT built from the set $ω$. For each $0\leq α\leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_ω$ is empty, as well as the limiting distribution of entropy for $X_ω$. For $α$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_ω$ contains a unique irreducible component of positi…
We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts they define are not topologically mixing. We then show that they satisfy a weaker mixing property using a numeration system arising from a sequence of lengths of inflated words. Moreover, we provide explicit bounds for the corresponding topological entropy in terms of the defining parameters $n$ and $p$.
The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the {\it word entropy} $E_W(f)$ associated to a given f…
In this note we prove the following results: $\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT, then $G$ has decidable word problem. More generally, for f.g. groups that are not recursively presented, there exists a computable obstruction for them to admit strongly aperiodic SFTs. $\bullet$ On the positive side, we build strongly aperiodic SFTs on some new classes of groups. We show in particular that some particular monster groups admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of group $G$, we show how to build strongly aperiodic SFTs ove…
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative fixed point f^ω(α), where α is in A. Consider the subshift (X, T), where X is the shift orbit closure of f^ω(α) and T: X --> X is the shift map. Let S be a finite alphabet that is in bijective correspondence via a mapping c with the set of nonempty suffixes of the images f(a) for a in A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0} such that π(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is primitive and f(A) is a suffix code, then there exists a mapping H: calS --> calS…
Computability of Pressure for Subshifts on Countable Amenable Groups
本文证明了可数顺从群上子位移的拓扑压强可从上方计算，且对强不可约有限型位移完全可计算，推广了以往仅适用于 $\mathbb{Z}^d$-子位移的结果，并给出计算时间显式界。
A multifractal analysis for uniform level sets with constraints on word appearance
本文研究有限型子移位中一致水平集的u-维数，证明其与普通水平集相同，并推广到满足子词约束的子集，应用于Gibbs测度的多重分形分析。
On the entropies of subshifts of finite type on countable amenable groups
本文引入群图工具，将可数 amenable 群上的子移位相互嵌入，并利用熵加法公式证明在容许类平移作用的有限生成群上，有限型子移位的拓扑熵集合与非负上半可计算实数集合一致，从而完整刻画了多类群中 SFT 的熵。
Equilibrium measures for two-sided shift spaces via dimension theory
该研究针对有限字母表上的双边移位空间与连续势函数，给出了用类Hausdorff测度构造描述平衡测度的条件，拓展了此前双曲系统的相关结果，覆盖有限型子移位、Hölder连续势及带同步字的移位空间。
Box dimensions of $(\times m, \times n)$-invariant sets
本文研究环面自同态不变集的盒维数，在非混合 sofic 情形下建立熵界，并给出编码子移位及非传递 sofic 情形的精确公式。
Symbol sequences from three-rotor coincidences and their word-complexity
通过记录三转子问题的配对和三次重合，提出四符号数字化方案，研究符号序列的词复杂度，发现混沌轨道词复杂度指数增长（拓扑熵log2），而准周期轨道词复杂度线性增长（零熵）。
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