Joshua Frisch, Omer Tamuz
A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.
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We discuss Devaney chaos on compact metric spaces using a decomposition space characterized by topological nature of symbolic dynamics. A chaotic map obtained here is defined as a topologically conjugate of the chaotic map on a decomposition space which is induced by a chaotic map of symbolic dynamics. In particular, the chaotic character of the tent map and the baker map on [0,1] are reconsidered based on decomposition dynamics involving symbolic dynamics with different two chaotic maps. As an example of compact metric space we exhibit a chaotic map existing on any given finite graph.
We develop some aspects of a general theory of presentations of subshifts by labelled directed graphs, in particular by compact graphs. Also considered are synchronization properties of subshifts that lead to presentations by countable graphs.
We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.
A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). The `dimension' dim(V) is an exponent describing the growth rate of balls in the digraph as a function of their radius. We show: if X has positive entropy and dim(V)>1, and the system (A^V,X,F) satisfies minimal symmetry and mixing condition…
Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems
本文研究非自治符号与膨胀动力系统，简化非自治符号系统中压力公式、推导非自治伯努利测度下测度理论压力公式并得到平衡态，还证明非自治膨胀系统有生成元的充要条件及强一致膨胀系统的等半共轭性质。
Meyer sets, Pisot numbers, and self-similarity in symbolic dynamical systems
本文从算术与动力学视角探讨准晶与自相似性的联系，重点阐述Pisot数在定义简单模型集及作为Meyer集伸缩因子中的作用，并介绍Yves Meyer关于Pisot或Salem数保持Meyer集的刻画。
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…
The symbolic complexity of an infinite word $W$ is the function $p_W(l)$ counting the number of different subwords in $W$ of length $l$. In this paper our main purpose is to study the complexity for a class of topological dynamical systems, called iceberg systems, given by the following symbolic procedure. Starting from a given finite word $w_1$ we construct a sequence of words $w_{n+1} = w_n ρ_{a_n(1)}(w_n)...ρ_{a_n(q_n-1)}(w_n)$, where $ρ_a(u)$ is the cyclic rotations of the word $u$ by $a$ positions, and consider an infinite word $W$ extending each $w_n$ to the right. It is shown that for i…
We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of entropies of (standard) full shifts. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical system.
This study investigates the natural or intrinsic measure of a symbolic dynamical system $Σ$. The measure $μ([i_{1},i_{2},...,i_{n}])$ of a pattern $[i_{1},i_{2},...,i_{n}]$ in $Σ$ is an asymptotic ratio of $[i_{1},i_{2},...,i_{n}]$, which arises in all patterns of length $n$ within very long patterns, such that in a typical long pattern, the pattern $[i_{1},i_{2},...,i_{n}]$ appears with frequency $μ([i_{1},i_{2},...,i_{n}])$. When $Σ=Σ(A)$ is a shift of finite type and $A$ is an irreducible $N\times N$ non-negative matrix, the measure $μ$ is the Parry measure. $μ$ is ergodic with maximum entr…
We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any $n$-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic shift actions on languages, and we study their orbitals and transversals. We present explicit formulations of the generating functions that allow us to deduce the orbital and transversal dimensions of classes of musical scales generated by vertex shifts, for all $n$, in particular for the 12-TET tuning system.
We associate to each discrete partial dynamical system a universal C*-algebra generated by partial isometries satisfying relations given by a Boolean algebra connected to the discrete partial dynamical system in question. We show that for symbolic dynamical systems like one-sided and two-sided shift spaces and topological Markov chains with an arbitrary state space the C*-algebras usually associated to them, can be obtained in this way. As a consequence of this, we will be able to show that for two-sided shift spaces having a certain property, the crossed product of the two-sided shift space i…
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace $X$ of $\mathcal{H}$ and its dual space $X'$. The three topological spaces $X \subset \mathcal{H} \subset X'$ is called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron-Frobenius operators of the one-sided and two-sided shifts of finite types (symbolic dynamical systems)…
To a given Pisot unit $β$ we associate a finite abelian group whose size appears to be equal to the discriminant of $β$. We call it the Pisot group and find its representation in the two-sided $β$-compactum in the case of $β$ satisfying the relation $Fin(β)=\Bbb Z[β]\cap[0,1)$. As a motivation for the definition, we show that the Pisot group is the kernel of some important arithmetic coding of the toral automorphism given by the companion matrix naturally associated with $β$.
The commonly used spatial entropy $h_{r}(\mathcal{U})$ of the multi-dimensional shift space $\mathcal{U}$ is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space $\mathbb{Z}^{d}$, $d\geq 2$. This work studies spatial entropy $h_Ω(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding system $Ω=\{Ω(n)\}_{n=1}^{\infty}$ where $Ω(n)$ is increasing finite sublattices and expands to $\mathbb{Z}^{d}$. $Ω$ is called genuinely $d$-dimensional if $Ω(n)$ contains no lower-dimensional part whose size is comparable to that of its $d$…
Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the stepped line associated with an element of the symbolic system on the contracting space of the matrix. We show that this Rauzy fractal depends continuously on the sequence of substitutions, and investigate some of its properties; in some cases, this construction gives a geometric model for the symbolic dynamical system.
This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by substitutional system, and dynamical properties are considered as criteria of superword being generated by interval exchange transformation. As a consequence, one can…
Typical periodic optimization for dynamical systems: symbolic dynamics
本文提出了一种在符号动力学中研究典型周期优化问题的新理论，证明了在特定条件下最大化测度的周期性及非周期性优化的结构性定理。
Analyzing Topological Mixing and Chaos on Continua with Symbolic Dynamics
本文提出一种分析机制，通过与传统符号动力学范畴外的定性动力系统共轭，证明了连续统上离散动力系统的拓扑混合与混沌的存在性，并在经典模型上验证了该方法。
Often in the study the periodic orbits in dynamical systems, the computation of the Lyapunov Coeficients is needed. In this paper, the calculations of this coeficients were done via complex variable transformation in order to obtain the complex normal form for a planar Bautin bifurcation. Some examples are given in order to verify the consistency of the algorithms.
This paper aims to initialize a dynamical aspect of symbolic integration by studying stability problems in differential fields. We present some basic properties of stable elementary functions and D-finite power series that enable us to characterize three special families of stable elementary functions involving rational functions, logarithmic functions, and exponential functions. Some problems for future studies are proposed towards deeper dynamical studies in differential and difference algebra.
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-simi…
Automorphisms of Locally Compact Groups, Symbolic Dynamics and the Scale Function
本文提出用符号动力系统对全不连通局部紧群的任意自同构建模，揭示了本质游荡可数状态Markov移位对尺度函数的独有影响，并证明了Haar测度下的传递性或遍历性排除了可数离散或本质游荡分量。
We numerically test the method of non-sequential recursive pair substitutions to estimate the entropy of an ergodic source. We compare its performance with other classical methods to estimate the entropy (empirical frequencies, return times, Lyapunov exponent). We considered as a benchmark for the methods several systems with different statistical properties: renewal processes, dynamical systems provided and not provided with a Markov partition, slow or fast decay of correlations. Most experiments are supported by rigorous mathematical results, which are explained in the paper.
A $C^*$-symbolic dynamical system $({\cal A}, ρ, Σ)$ is a finite family $\{ρ_α\}_{α\inΣ}$ of endomorphisms of a $C^*$-algebra ${\cal A}$ with some conditions. It yields a $C^*$-algebra ${\cal O}_ρ$ from an associated Hilbert $C^*$-bimodule. In this paper, we will extend the notion of $C^*$-symbolic dynamical system to $C^*$-textile dynamical system $({\cal A}, ρ, η, {Σ^ρ}, {Σ^η}, κ)$ which consists of two $C^*$-symbolic dynamical systems $({\cal A}, ρ, {Σ^ρ})$ and $({\cal A}, η, {Σ^η})$ with certain commutation relations $κ$ between their endomorphisms $\{ρ_α\}_{α\in Σ^ρ}$ and $\{η_a \}_{a \in…
We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is giv…
We consider the dynamical behavior of Martin-Löf random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-S…
New Symbolic Procedures in the Study of Dynamical Systems
本论文在有限维交换实代数中发展动力系统的符号表示，利用代数框架支持傅里叶、拉普拉斯变换及沃尔什函数辨识，为非惯性系运动提供精确向量解，并重建1995年原版。
These notes from lectures by the author at the CIMPA Research School held in Salta, Argentina, in November 2015 review definitions and properties of various measures of complexity in topological and measure-preserving dynamical systems, especially symbolic dynamical systems.
Approximations of symbolic substitution systems in one dimension
该研究针对准晶体薛定谔算子谱分析需求，研究一维替换动力系统的周期逼近，给出逼近子系统候选、收敛性刻画与速率估计，并应用于示例，为准晶体相关算子谱分析提供工具。
WE trace the beginning of symbolic dynamics--the study of the shift dynamical system--as it arose from the use of coding to study recurrence and transitivity properties of geodesics. The normal citations for the first appearance of symbolic dynamics, Hadamard's 1898 paper and the 1938 and 1940 papers of Morse and Hedlund, don't truly represent the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the beginning of symbolic dynamics in a 1944 paper by Hedlund.
In this paper we propose a new symbolic-numeric algorithm to find positive equilibria of a n-dimensional dynamical system. This algorithm implies a symbolic manipulation of ODE in order to give a local approximation of differential equations with power-law dynamics (S-systems). A numerical calculus is then needed to converge towards an equilibrium, giving at the same time a S-system approximating the initial system around this equilibrium. This algorithm is applied to a real biological example in 14 dimensions which is a subsystem of a metabolic pathway in Arabidopsis Thaliana.
This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the cir…
Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction
本文提出几乎线性循环神经网络（AL-RNNs），可从时间序列数据中自动学习动力系统的最简约分段线性表示，并产生保留拓扑性质的符号编码，在Lorenz、Rössler系统及经验数据上验证了其可解释性与有效性。
In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various properties of such groups and establish in particular a reversibility and invertibility theorem for injective endomorphisms of symbolic group varieties over weakly surjunctive group universes with algebraic group alphabets in arbitrary characteristic. We also obtain novel evidence related to Kaplansky's stable finiteness conjecture.
We define a generic algorithmic framework to prove pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi-Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamic…
Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian…
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