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.
189 questions
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Chaotic dynamics of maps on unit square that are NOT Triangular
We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
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Conjugacy between piecewise linear circle maps
Let $\mathcal{M}$ the Mandelbrot set,
$\mathcal{M}=\{c \in \mathbb{C}: \{Q_c^n(0) \}_{n \in \mathbb{N}} \text{ is bounded, where } Q_c(z)=z^2+c \}$
And let the hyperbolic or stable component, $H_n=\{ ...
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A Collatz-like map?
Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows:
Set $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
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Is the finite/countable union of topological Markov shifts a topological Markov shift?
A topological Markov shift (TMS), or countable state shift of finite type (SFT), is a shift space $X$ over a countably infinite alphabet $\mathcal{A}$ defined by a transition matrix $T=(t_{ij})_{\...
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Number of periodic points of subshift of finite type
Let $(X, \sigma)\subset (\{0, 1, 2, 3\}^\mathbb{N},\sigma)$ be a subshift of finite type. Let $P_n$ be the set of $n$-periodic points. If $|P_n|=2^n$ for all $n\ge 1$, then it is true that $(X, \sigma)...
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Computing the language of an $S$-adic shift
I have been looking online for how or if one can compute the language of an $S$-adic subshift generated by finitely many substitutions. I know that one can compute the language of a substitution ...
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Decidability of periodic tilings of the plane
I'm interested in tilings of the plane by squares, with labels on the edges. It's well known that (1) the question "can one tile the plane with the following finite set of tiles?" is undecidable, and (...
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Asymptotic growth rate for primitve S-adic systems
It is known that for a primitive substitution $S:\mathcal{A}\to \mathcal{A}^+$, there exists constants $c,C>0$ such that
$$ c\theta_S^n \leq \vert S^n(a)\vert \leq C \theta_S^n \quad \text{for all} ...
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Extension of automorphism of shift of finite type
$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two subshifts of finite type and $X\subset Y$, and $\phi:X\rightarrow X$ be a homeomorphism commuting with the shift map. Is there any homeomorphism $\...
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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 ...
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Question regarding characterization of linearly recurrent subshifts by Durand
I was looking at the following paper by Fabien Durand, Corrigendum and addendum to ‘Linearly recurrent subshifts have a finite number of non-periodic factors’.
I have somewhat of a basic question ...
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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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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 $\{ x_n \}$ ...
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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 ...
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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 $T_{1}M$ ...
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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 ...
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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 ...
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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}...
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Decidability of (restricted) periodicity of Wang tilings
Consider a Wang tiling (given a subset of $C^4$ for a finite set $C$ of colours, e.g.). It's well-known to be undecidable whether there exists a tiling, and also whether there exists a periodic tiling....
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What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
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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 ...
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Substitutions and Sturmian sequences
We know that any substitution can generate sequence, for example the Fibonacci substitution:
$\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of $\...
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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 ...
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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\}^...
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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 ...
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Fast algorithms for external angle computations
Two related problems related to the complex quadratic polynomial $f_c(z) = z^2 + c$ and Mandelbrot and/or Julia sets:
find an external angle $\theta_c$ for a complex point $c$
find a complex point $...
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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 ...
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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 ...
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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 ...
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Multi dimensional symbolic dynamics
I want to learn Multi dimensional symbolic dynamics. can you point to any recent thesis containing a good exposition or lecture notes?
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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 ...
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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,...,...
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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\...
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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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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 ...
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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 ...
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Are there uncountably many cube-free infinite binary words?
In Cube-free infinite binary words it was established that there are infinitely many cube-free infinite binary words (see the earlier question for definitions of terms). The construction given in ...
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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 ...
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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?
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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 ...
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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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How can generic closed geodesics on surfaces of negative curvature be constructed?
As far as I understand it the closing lemma implies that closed geodesics on surfaces of negative curvature are dense. So: how can they be constructed in general?
A concrete answer that dovetails with ...
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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'...
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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
...
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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 ...
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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 ...
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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 ...
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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 ...
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Extension of Coburn's theorem on isometry and Toeplitz algebra
$\newcommand{\id}{\mathrm{id}}$Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = \id$ and $XX^{\star} \neq \id$). Coburn's theorem states that ${\rm C}^{\star}(X)$, ...
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