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.

Filter by
Sorted by
Tagged with
3 votes
0 answers
56 views

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
  • 143
2 votes
1 answer
192 views

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
3 votes
1 answer
87 views

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)...
Bo Peng's user avatar
  • 31
2 votes
0 answers
82 views

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
6 votes
0 answers
86 views

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
2 votes
1 answer
105 views

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
2 votes
1 answer
145 views

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
1 vote
1 answer
131 views

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
1 vote
1 answer
63 views

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
2 votes
1 answer
77 views

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
0 votes
0 answers
107 views

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
2 votes
1 answer
117 views

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
0 votes
0 answers
78 views

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
3 votes
1 answer
184 views

'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
4 votes
2 answers
252 views

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
1 vote
1 answer
153 views

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 ...
Numbra's user avatar
  • 113
1 vote
2 answers
294 views

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
1 vote
1 answer
171 views

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
0 votes
1 answer
116 views

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
0 answers
318 views

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
307 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
3 votes
1 answer
115 views

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
  • 1,001
4 votes
1 answer
246 views

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\...
Anthony Quas's user avatar
  • 22.4k
1 vote
1 answer
130 views

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
  • 119
2 votes
1 answer
151 views

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
  • 1,001
1 vote
1 answer
187 views

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
  • 51
11 votes
1 answer
436 views

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 MacDonald's user avatar
6 votes
0 answers
151 views

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
71 views

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
0 votes
0 answers
65 views

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
0 votes
1 answer
137 views

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
  • 159
4 votes
0 answers
93 views

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
136 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
108 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
0 votes
0 answers
52 views

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
  • 16.6k
14 votes
2 answers
888 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
171 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
  • 123
2 votes
1 answer
204 views

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
  • 1,001
2 votes
0 answers
112 views

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
73 views

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
  • 101
1 vote
0 answers
176 views

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
  • 111
1 vote
0 answers
61 views

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
  • 323
1 vote
0 answers
54 views

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
0 answers
148 views

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
633 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
  • 1,001
6 votes
1 answer
243 views

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
  • 1,001
6 votes
0 answers
135 views

Difficulty of homeomorphism of effective Cantor dynamics

Let $X = \{0,1\}^{\mathbb{N}}$ with the product topology. Given a Turing machine $M$ and $x \in X$, define $M(x) \in \{0,1\}^* \cup X$ as the sequence of bits output by $M$ when given an oracle for $x$...
Ville Salo's user avatar
  • 6,337
1 vote
1 answer
93 views

Explicit transitive flow on disc

$D_n\triangleq \left\{x \in \mathbb{R}^n:\, \|x\|\leq 1\right\}$ with its subspace topology. By a transitive flow on $D_n$ I mean a continuous function $$ \phi: [0,1]\times D_n\rightarrow D_n, $$ ...
ABIM's user avatar
  • 5,119