Nick Ramsey
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 $S_i$.
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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.
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.
Subshifts of finite symbolic rank
本文受遍历理论启发研究有限符号秩子移位，证明其与有限拓扑秩极小Cantor系统的关系，刻画秩1子移位共轭类的稠密性与非通有性，还探讨了其拓扑因子性质。
A necessary and sufficient condition is given for the existence of an embedding of an irreducible subshift of finite type into the Fibonacci-Dyck shift
We prove in this text a characterization of the possible entropy dimensions of minimal tridimensional subshifts of finite type with a computability condition, using Goldbach's theorem on Fermat numbers.
In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.
In this note we prove that Moebius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all $C^{1+α}$ surface diffeomorphisms with positive entropy correlate with the Moebius function.
We prove an escape rate result for special semi-flows over non-invertible subshifts of finite type. Our proofs are based on a discretisation of the flow and an application of an escape rate result for conformal repellers.
Domino tilings have been studied extensively for both their statistical properties and their dynamical properties. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results about domino tilings to show that our subshift of finite type has a measure of maximal entropy with which the subshift has completely positive entropy but is not isomorphic to a Bernoulli shift.
We study equidistribution of certain subsets of periodic orbits for subshifts of finite type. Our results solely rely on the growth of these subsets. As a consequence, effective equidistribution results are obtained for both hyperbolic diffeomorphisms and expanding maps on compact manifolds.
We prove that a subshift $(X,T)$ is linearly recurrent if and only if it is a primitive and proper $S$-adic subshift. This corrects Proposition 6 in F. Durand ({\it Ergod. Th. & Dynam. Sys. {\bf 20}} (2000), 1061--1078).
We introduce subshifts of quasi-finite type as a generalization of the well-known subshifts of finite type. This generalization is much less rigid and therefore contains the symbolic dynamics of many non-uniform systems, e.g., piecewise monotonic maps of the interval with positive entropy. Yet many properties remain: existence of finitely many ergodic invariant probabilities of maximum entropy; lots of periodic points; meromorphic extension of the Artin-Mazur zeta function.
This paper studies several aspects of symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $π^{-1}(y)$ of factor maps $π\colon(X,T)\to(Y,T)$ between minimal $\mathcal{S}$-adic subshifts of finite alp…
This paper studies several aspects of symbolic ({\em i.e.}\ subshift) factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $π^{-1}(y)$ of factor maps $π\colon(X,T)\to(Y,S)$ between minimal $\mathcal{S}$-adic …
We define the finite extension property for $d$-dimensional subshifts, which generalizes the topological strong spatial mixing condition defined by Briceño (2016), and we prove that this property is invariant under topological conjugacy. Moreover, we prove that for every $d$, every $d$-dimensional block gluing subshift factors onto every $d$-dimensional subshift which has strictly lower entropy, a fixed point, and the finite extension property. This result extends a theorem from Boyle, Pavlov, and Schraudner (2010), which requires that the factor contain a safe symbol.
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.
Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic subshifts. This is done establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like all…
For each $n, d \in \mathbb{N}$ and $0 < α< 1$, we define a random subset of $\mathcal{A}^{\{1, 2, \dots, n\}^d}$ by independently including each element with probability $α$ and excluding it with probability $1-α$, and consider the associated random subshift of finite type. Extending results of McGoff and of McGoff and Pavlov, we prove there exists $α_0 = α(d, |\mathcal{A}|) > 0$ such that for $α< α_0$ and with probability tending to $1$ as $n \to \infty$, this random subshift will contain only finitely many elements. In the case $d = 1$, we obtain the best possible such $α_0$, $1/|\mathcal{A}…
We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. In case of Markov measures we give a further example of St.\ Petersburg type distribution functions. To prove these statements we introduce the space of quasi-Hölder continuous functions for subshifts of finite type.
In this paper we consider the structure of $ω$-limit sets in subshifts of Baire space. We consider both subshifts of finite type and subshifts of bounded type and we demonstrate that many classical structure theorems for $ω$-limit sets fail in this context. Nevertheless, we obtain characterizations of $ω$-limit sets in subshift of finite types and of attracting $ω$-limit sets in subshifts of bounded type.
We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily finite. We also prove that a subshift over any finitely generated group that consists of finite orbits is finite, and related results for tilings of Euclidean space.
Homomorphisms from aperiodic subshifts to subshifts with the finite extension property
本文证明在群子群多项式增长、源子移位非周期且目标子移位具有有限扩展性质的条件下，存在同态，并结合熵条件与无全局周期条件实现嵌入。
Let $X\subset A^{Z^d}$ be a $2$-dimensional subshift of finite type. We prove that any $2$-dimensional multidimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general $d$-dimensional case. We prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. In the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. We also give sufficient conditions for the shift space $X$ to…
In this paper we study the quenched distributions of hitting times for a class of random dynamical systems. We prove that hitting times to dynamically defined cylinders converge to a Poisson point process under the law of random equivariant measures with super-polynomial decay of correlations. In particular, we apply our results to uniformly aperiodic random subshifts of finite type equipped with random invariant Gibbs measures. We emphasize that we make no assumptions about the mixing property of the marginal measure.
Topo-isomorphisms of irregular Toeplitz subshifts for residually finite groups
针对每个可数剩余有限群，构造了不规则Toeplitz子移位作为极大等连续因子的拓扑同构扩展，并证明了相关充分条件及非空性，为顺从群提供了非正则平均拟连续扩展。
We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class …
Consider a Hölder continuous potential $ϕ$ defined on the full shift $A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $μ_ϕ$ on $X$ associated to $ϕ$. Besides, there is a natural nested sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift $X$. To this sequence we can associate a sequence of Gibbs measures $(μ_ϕ^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $μ_ϕ$ (in the classical distance metrizing weak topology). We also establish…
R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics, correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it, and to expose some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way of the pos…
This paper establishes topological (equi-)semiconjugacy and (equi-)conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li-Yorke chaos and distributional chaos in a sequenc…
We prove that a shift ergodic measure on a topologically mixing sub-shift is isomorphic to a Bernoulli shift whenever it is quasi invariant under permutations of finite number of coordinates. We prove also that Gibbs measures on topologically mixing subshift of finite type are quasi invariant.
Geometrical representation of subshifts for primitive substitutions
本文研究 Perron 特征值为 Pisot 单位的本原替换的子移位，构造了可测共轭的域交换，给出环面平移有限扩张条件，证明弱不可约 Pisot 替换子移位性质，并提供计算伪幺模替换子移位特征值的算法。
On surjunctive and injunctive subshifts of finite type
本文通过不可约分量和Cantor-Bendixson分解刻画了Z-有限型子转移的满射性和单射性，证明了满射性等价于Moore性质，且单射性蕴含满射性。
An embedding theorem for multidimensional subshifts
本文针对任意可数阿贝尔群上的有限型子移位提出新的嵌入定理，给出任意子移位嵌入目标子移位的充分必要条件，首次完整刻画多维全移位的子系统，并将 Boyle 的低熵因子定理推广到多维情形。
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 …
In this text, we prove then that any minimal effective dynamical system on a Cantor set $\mathcal{A}^{\mathbb{N}}$ can be simulated by a minimal $\mathbb{Z}^3$-SFT, in a sense that we explicit here. This notion is a generalization of simulation by factor and sub-action defined by Hochman.
We discuss a method of calculating the zeta function of subshifts which have a presentation by a finite directed graph labeled by elements of the associated inverse semigroup. This class of subshifts is introduced as a class of property A subshifts(T.Hamachi, K.Inoue and W.Krieger/Subsystems of finite type and semigroup invariants of subshifts, preprint), and the Dyck shift and the Motzkin shift are representative subshifts in this class. The exact number of the periodic points and entropies are also given by this method, and these values are used in the embedding condition for an irreducible …
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。
\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate …
In their recent paper [KNS2019], the first author, S. Kiriki, and T. Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In this paper we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any Hölder continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large …
On the stability of the penalty function for the $\mathbb{Z}^2$-hard square shift
本文研究二维有限型子平移惩罚函数最大化测度的稳定性问题，针对正拓扑熵情形，证明满足单点可填充性质的最近邻有限型子平移的惩罚函数在Lipschitz扰动下保持稳定，与Robinson平铺的不稳定性形成对比。
On the structure of the Birkhoff-irregular set for subshifts of finite type
针对拓扑混合有限型子移位，本文证明 Birkhoff 不规则集包含不可数个两两不交、稠密且具有全拓扑熵与全 Hausdorff 维数的不变子集。
Parametrization by Horizontal Constraints in the Study of Algorithmic Properties of $\mathbb{Z}^2$-Subshift of Finite Type
本文通过预先固定水平约束作为参数，研究$\mathbb{Z}^2$-有限型子位移的多米诺问题不可判定性条件及熵可实现的右递归可枚举数范围，并区分任意局部规则与仅垂直规则两种情形。
For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple, and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.
We show the Williams Conjecture is false for irreducible shifts of finite type by examining relative sign-gyration numbers of conjugacies between shifts with no points of period one or two.
We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.
On the minimal components of substitution subshifts
本文研究了替换移位的最小成分，精确刻画了其结构并提供了算法计算方法。
In this paper, we provide an effective method to compute the topological entropies of $G$-subshifts of finite type ($G$-SFTs) with $G=F_{d}$ and $S_{d}$, the free group and free semigroup with $d$ generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of $S_{2}$-SFTs, namely the types $\mathbf{E},\mathbf{D},\mathbf{C}$ and $\mathbf{O}$, are introduced and we could compute their entropies explicitly. This enables us to give the complete characterization of $S_{2}$-SFTs on two symbols. That is, …
$λ$-graph systems are labeled Bratteli diagram with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of $λ$-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems are $G$-strong shift equivalent if and only if their presented subshifts are $G$-conjugate. $G$-equivalent classes of subshifts are classified by the cohomology classes of their associated skewing functions.
Periodic points and residual finiteness of automorphism groups of subshifts
研究子移位的周期点密度与其自同构群余有限性之间的关系，证明在周期点不密集时，自同构群非余有限，并构造了一个非余有限的强不可约子移位例子。
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.
In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction...). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d -- that is a subshift whose set of forbidden patterns can be generated by a Turing machine -- can be obtained by applying dynamical operations on a subshift of finite type of dimension d + 1 -- a subshift that can…
In this paper we study the directions of periodicity of three-dimensional subshifts of finite type (SFTs) and in particular their slopes. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope being the angle of the periodicity vectors. In this paper, we prove that any $Σ^0_2$ set may be realized as a a set of slopes of an SFT.
Let $G$ be a residually finite group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in $X$. The density of periodic configurations implies in particular that every injective endomorphism of $X$ is surjective and that the group of automorphisms of $X$ is residually finite. We also introduce a class of subshifts $X \subset A^\Z$, including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense.
Toeplitz subshifts of finite rank
本文研究有限拓扑秩 Toeplitz 子移位，通过构造反例否定刻画问题并证明强 Toeplitz 子移位集合的通有性，同时确定拓扑秩 2 情形下共轭等分类问题的复杂度，并刻画其自同构群结构。
We give strongly aperiodic subshifts of finite type on every hyperbolic surface group; more generally, for each pair of expansive primitive symbolic substitution systems with incommensurate growth rates, we construct strongly aperiodic subshifts of finite type on their orbit graphs.
Let ${\mathcal P}\subset{\mathbb Z}^2$ be a convex polygon with each vertex in it labeled by an element from a finite set and such that the labeling of each vertex $v\in {\mathcal P}$ is uniquely determined by the labeling of all other points in the polygon. We introduce a class of ${\mathbb Z}^2$-shift systems, the {\em polygonal shifts}, determined by such a polygon: these are shift systems such that the restriction of any $x\in X$ to some polygon ${\mathcal P}$ has this property. These polygonal systems are related to various well studied classes of shift systems, including subshifts of fin…
In this paper, we study the behaviour of the longest common substring for random subshifts of finite type (for dynamicists) or of the longest common substring for random sequences in random environments (for probabilists). We prove that, under some exponential mixing assumptions, this behaviour is linked to the Rényi entropy of the stationary measure. We emphasize that what we establish is a quenched result.
For a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\text{Aut}^{(\infty)}(T)$ consists of all self-homeomorphisms of $X$ which commute with some power of $T$. Motivated by the study of these groups in the context of shifts of finite type, we introduce a certain entropy for groups called local $\mathcal{P}$ entropy. We show that when $(X,T)$ is a non-trivial mixing shift of finite type, the local $\mathcal{P}$ entropy of the group $\text{Aut}^{(\infty)}(T)$ is determined by the topological entropy of $(X,T)$. We use this to give a complete cl…
We prove inequalities relating the measures of maximal entropy of two patterns u,v where the extender set of u is contained in the extender set of v. Our main results are two generalizations of a Theorem of Meyerovitch; the first applies to all such v,w when G=Z, and the second to v,w with the same shape and any countable amenable finitely generated torsion-free G. As a consequence of our results we give new and simpler proofs of several facts about synchronizing subshifts and we answer a question of Climenhaga.
Aperiodic subshifts of finite type on groups which are not finitely generated
该研究给出了一个存在非空强非周期有限型子移位的非有限生成群实例，并通过群的有限生成子群与共轭类的根完整刻画了具备该性质的群的特征。
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