Questions tagged [symbolic-dynamics]

Symbolic dynamics is the study of dynamical systems defined in terms of shift transformations on spaces of sequences. Examples of topics in this area include shifts of finite type, sofic shifts, Toeplitz shifts, Markov partitions and symbolic coding of dynamical systems.

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Estimate for the length of a partial orbit for a shift map for which its delta neighbourhood covers an interval

Consider $f:[0,2\pi) \to [0,2\pi )$ given by $f(x) = (x + 1) \bmod 2\pi$ for all $x\in [0,2\pi )$, i.e. a shift map on the unit circle with anti-clockwise shift of $1$. Denote the sequence $\{ x_n \}$ ...
JCM's user avatar
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Properties of limit set for cellular automata

Is anyone familiar with results about properties of the limit set of the local rule for a cellular automaton? I haven't been able to find any good materials on the subject from an initial search, and ...
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Second eigenvalue of primitive matrix

Let $A$ be a primitive $N\times N$-matrix with positive entries, that is there is $n>0$ such that $(A^n)_{i,j}>0$ for all $i,j$. For brevity, assume the entries consist only of $0$ and $1$. The ...
Curious's user avatar
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Invariant measure of geodesic flow on unit tangent bundle of a modular surface

This is a paper written by Series "THE MODULAR SURFACE AND CONTINUED FRACTIONS". I want to know about above construction natural invariant measure $\mu$ for the geodesic flow on $T_{1}M$ ...
user473085's user avatar
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Extending isomorphism between subsystems in shift system

Let $(\Sigma^{\mathbb{Z}},S)$ be a left-shift system, where $\Sigma$ is a metrizable compact set. Consider the automorphism group of it (bijective factor maps of itself), denoted by $G$. Now let $(A,S)...
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Aperiodic SFT equal to a substitution subshift

I was wondering whether there are primitive symbolic substitutions over $\mathbb{Z}^d$ and alphabet $\mathcal{A}$ whose associated subshift is equal to an aperiodic SFT. By SFT here I mean a subshift ...
Keen-ameteur's user avatar
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Automorphism groups of subshifts and factor maps

Let $\pi : X \to Y$ be a factor map between subshifts over finite alphabets. Let $\operatorname{Aut}(X)$ and $\operatorname{Aut}(Y)$ stand for automorphism groups of these shifts. We say that $\varphi ...
Dominik Kwietniak's user avatar
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Morse-Hedlund\Coven-Hedlund theorem for non-Abelian groups

There is a well know theorem by Coven and Hedlund, in Sequences with minimal block growth, stating that the complexity function of an aperiodic sequence\configuration $\omega\in \mathcal{A}^{\mathbb{Z}...
Keen-ameteur's user avatar
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A sensitive 2-dimensional cellular automaton with a blocking word

I'am a Ph.D student in the domain of discrete dynamical systems. My thesis is about spectral properties of cellular automata in higher dimension. Kurka gives a classification for one dimensional ...
Nassima AIT SADI's user avatar
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Approximation of subshifts in Hausdorff distance

I have recently been interested in some questions which stem from taking subshifts which converge to a limiting subshift in the Hausdorff metric. More specifically, given an alphabet $\mathcal{A}$, I ...
Keen-ameteur's user avatar
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Computing admissible patches of a substitution

I have been recently trying to look at substitution tilings with finite local complexity by examining their admissible patch\pattern atlas, which is sometimes called their language. I have also seen ...
Keen-ameteur's user avatar
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Lower bounds for pattern complexity of linearly repetitive aperiodic subshifts

I recently asked in this thread about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c_n(\Omega)$ as the number of possible patterns on $Q_n=\{0,...,n−1\}^...
Keen-ameteur's user avatar
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Growing gliders under rule 110

I found a glider in the evolution space of rule 110 that grows constantly in size. Normal gliders live in the so-called ether, e.g. the so-called E-glider: Other – often complex – gliders exist in an ...
Hans-Peter Stricker's user avatar
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Reference on relation between SFTs and Wang-tiles

I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a ...
Keen-ameteur's user avatar
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Relation between symbolic substitution and cellular automata

I recently asked this on Math Stackexchange recently in this thread. I was told that there is a relation between symbolic substitutions and cellular automata. I'm vaguely familiar with Cobham's ...
Keen-ameteur's user avatar
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'Trivial' lower bounds for pattern complexity of aperiodic subshifts

I recently asked in this thread about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c_n(\Omega)$ as the number of possible patterns on $Q_n= \big\{ 0,...,...
Keen-ameteur's user avatar
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Lower bounds for pattern complexity of aperiodic subshifts

In the setting of symbolic dynamics over $\mathbb{Z}^d$, one can define for the $n$-th pattern complexity of a given a subshift $\Omega\subseteq \mathcal{A}^{\mathbb{Z}^d}$ as $$ c_n(\Omega):= \Big\...
Keen-ameteur's user avatar
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Topological full groups of subshifts: differences between one-dimensional and multi-dimensional subshifts

For a multidimensional subshift $X$ over $\mathbb Z^d$, the topological full group $[X]$ is the set of homeomorphisms $f$ of $X$ that can be written as $f : x \mapsto \sigma_{c(x)}(x)$ with $c : X \to ...
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Sufficient conditions for periodic tiling by Wang tiles

I'm recently interested in whether a sub-shift of finite type contains a doubly-periodic problem, when the set of configurations is of the sort $\mathcal{A}^{\mathbb{Z}^2}$. When $Q_2=\{0,1\}^2$, and ...
Keen-ameteur's user avatar
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Possible weaker version of the Domino/Wang tiling problem

This may be a dumb question, but I was wondering whether the question of 'periodically tiling the plane from a finite set of tiles' is the same as the domino tiling problem or a weaker version. I ...
Keen-ameteur's user avatar
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A special kind of pseudo-garden eden states in cellular automata

I'm currently investigating Wolfram's elementary cellular automata on finite grids with periodic boundary conditions, i.e. on $\mathbb{Z}/k$ for different $k$. It is clear that for each rule $R$ and ...
Hans-Peter Stricker's user avatar
6 votes
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Examples of expansive homeomorphisms with the specification property that are neither symbolic nor factors of mixing SFT nor product of thereof

I am looking for nontrivial examples of expansive homeomorphisms with the specification property on compact metric spaces. Here, by a ``trivial'' example I understand a subshift with the specification ...
Dominik Kwietniak's user avatar
6 votes
2 answers
309 views

Topological dynamical systems with only zero-entropy factors

Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?
user119197's user avatar
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the definition of the topological pressure for matrices

Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma_n$ the set of addmisible words with the ...
Adam's user avatar
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Word combinatorics terminology question

I'm looking for the name of what I suspect must be a standard property, and also for a possible statement about that property. First the property: $W=a_0\ldots a_{n-1}$ has this property if for all $1\...
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Properties of Følner sequences for countably infinite, finitely generated, amenable, periodic/torsion groups

I've managed to prove certain things about a class of groups, and the only remaining class of groups are those specified in the title. I'm mainly studying symbolic dynamics and not group theory, so I'...
Jacob R's user avatar
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1 answer
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Union of admissible words are subshift of finite type

Assume that $Q=(q_{ij})$ is a $k\times k$ with $q_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma_{Q}\rightarrow \Sigma_{Q}$, where ...
Adam's user avatar
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Proof that Sturmian shift is uniquely ergodic using irrational rotation

I am finding proofs of unique ergodicity of Sturmian shifts however I want to know if there is a proof that link that to the unique ergodicity of irrational rotations through conjugacy for example or ...
kiki 's user avatar
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Cohomology for extension problems in symbolic/topological dynamics?

Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
Sophie M's user avatar
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Construction of minimal zero entropy measure-theoretically strong mixing subshift?

Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is (1) minimal (2) zero (topological) entropy (3) measure-theoretically strong mixing (for some measure)? I am in particular ...
Ronnie Pavlov's user avatar
1 vote
1 answer
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Computing kneading sequences for renormalizations of Lorenz maps

I am stuck trying to understand certain claims made in this paper, and for completeness I will reproduce some definitions from it. A Lorenz map $f$ on $I = [0,1]$ is a monotone increasing function ...
user482093's user avatar
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Show that two matrices are strongly shift equivalent

The following question is from Introduction to dynamical systems, written by Michael Brin and Garrett Stuclk. Given two non-negative integer square matrices $A, B$, we say $A, B$ are elementarily ...
Sanae Kochiya's user avatar
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Why all the coefficients of the center manifold of this system are zeros?

I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
Mr. Proof's user avatar
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String rewrite system for algebraic knots/links?

$\newcommand\over{\vert}\newcommand\rot[1]{\mathopen<#1\mathclose>}$By its definition, an algebraic tangle, and by extension, its closure (knot or link) can be written as a string (of ...
Hauke Reddmann's user avatar
6 votes
1 answer
137 views

Subshifts with special property

I am looking how to prove the following fact: If $ X \subseteq A^\mathbb{Z}$ is an infinite minimal subshift, then for any $N\ge 1$, $X$ is conjugate to a minimal subshift $Y\subseteq B^\mathbb{Z}$ ...
Mustafa Gokhan Benli's user avatar
3 votes
0 answers
109 views

Description of Anderson-Putnam CW-complex construction

I have been trying to read the paper, Topological invariants fo substitution tiling and their associated $C^*$-algebras, to learn more about a construction of Anderson-Putnam complexes. However, it ...
Keen-ameteur's user avatar
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Statistical characteristics of low complexity subshifts

I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
R W's user avatar
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14 votes
2 answers
896 views

Open problems in symbolic dynamics

I would like to know which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. ...
2 votes
1 answer
172 views

Exponential mixing for subshifts

I asked this question on Math.StackExchange some time ago and got no responses. Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type $$ \...
QMath's user avatar
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2 votes
1 answer
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Irrational rotations are rank 2 by intervals without spacers

Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$. S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
Dominik Kwietniak's user avatar
3 votes
1 answer
165 views

Does full shift have the local product structure?

We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is ...
Adam's user avatar
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2 votes
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Birth of chaos due to nonautonomous perturbation

Let $\sigma, b>0$. I want to study the dynamics of the map $$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that $$T_{\sigma,b}(n,\theta,y) = (\...
Giuseppe Tenaglia's user avatar
2 votes
0 answers
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When is replacing the prefix of an angled internal address a valid operation?

While working on an artwork exploring patterns in the Mandelbrot set fractal, I constructed an angled internal address by: $$ 1 \overset{1/2}\longrightarrow 2 \overset{1/2}\longrightarrow 3 \overset{1/...
Claude's user avatar
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Is there a condition for a subshift of finite type to be uniquely ergodic?

Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic?
otah007's user avatar
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Number of permitted words up to permutation in a subshift

Let $A$ be a finite set and let $X \subseteq A^{\mathbb{N}}$ be a subshift. Let $\mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w \in \mathcal{L}_n$, one can ...
Adam's user avatar
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Continuity of Kneading invariants of generalised $\beta$-trasformations

For $\beta \in (1,2]$ and $\alpha \in [0,2-\beta]$ consider the generalised $\beta$-transformation $T_{\alpha,\beta}:[0,1] \to [0,1]$ to be $$T_{\alpha, \beta}(x) = \beta x + \alpha \mod 1.$$ It is a ...
Rafael Alcaraz Barrera's user avatar
1 vote
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Is a set over which dynamics are topologically conjugate to a shift map on two symbols always repelling?

Consider the one-sided full shift map $\sigma$ and the associated shift space of infinite sequences in two letters $\{0,1\}^\mathbb{N}$ on which the shift map acts, equipped with the usual metric. ...
aghostinthefigures's user avatar
2 votes
3 answers
634 views

The critical exponent function

It is a known fact [1] that, for every $c\in (1,\infty]$, it is possible to find a finite alphabet $\mathcal{A}$ and a word $w\in \mathcal{A}^\omega$ such that $w$ has critical exponent $c$. It looks ...
Alessandro Della Corte's user avatar
1 vote
1 answer
106 views

Lyapunov spectrum($h_{\mathrm{top}}(K(\alpha))$) achieves a positive value somewhere

$\DeclareMathOperator{\top}{\mathrm{top}}$Let $(\Sigma, T)$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Hölder continuous map. Let $$K(\alpha)=\Big\{x\in \...
Adam's user avatar
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6 votes
1 answer
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A unique equilibrium state which does not have Gibbs property

Let $T:\Sigma \rightarrow \Sigma$ be a topologically mixing subshift of finite type and let $f:\Sigma \rightarrow \mathbb{R}$ be a continuous functions over $(T, \Sigma)$. Assume that there is a ...
Adam's user avatar
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