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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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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
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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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177 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
639 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
107 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
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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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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
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1 answer
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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
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2 votes
1 answer
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Search for a general formula from known iterative relation

$F$ is a mapping among $\{\theta_{n_1n_2}\}$, with $\eta_{1/2}$ being arbitrary constants involved. $F: \theta_{n_1n_2} \rightarrow \theta_{n_1+1n_2}+\theta_{n_1n_2+1}+\eta_{1}n_1\theta_{n_{1}-1n_{2}} ...
dhem's user avatar
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Confusion on the assumption when discussing the kneading invariants for unimodal maps

A unimodal map is a continuous map $f:[0,1]\longrightarrow [0,1]$ such that there is only one turning point (critical point), denoted by $c$, and $f(0)=f(1)=0$. Unimodal map is related to kneading ...
JacobsonRadical's user avatar
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2 answers
374 views

Diophantine equation that has an infinite number of positive integers solutions

Let us consider a sequence of continuous functions $g_{q}:ℝ^2\to ℝ^2$. Let $(A_{q})_{q\geq 1}$ be a sequence of compact sets in $ℝ^2$. Assuming that each function $g_{q}$ is topologically mixing in $...
Safwane's user avatar
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Example of primitive substitution with two rationally independent eigenvalues?

I am looking for an example of a primitive substitution $\sigma$, not Pisot, such that the associated subshift $X_\sigma$ has two irrational and rationally independent eigenvalues. Equivalently, a ...
vertebra's user avatar
10 votes
0 answers
475 views

If $(Y,T)$ is a connected minimal system with a symbolic extension of linear word complexity, is $(Y,T)$ equicontinuous?

Let $(Y,S)$ be a minimal topological dynamical system such that $Y$ is connected. A simple example of a system like this is an irrational rotation of the circle, and it is known that Sturmian ...
user158448's user avatar
1 vote
2 answers
232 views

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 [......
M. Winter's user avatar
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Does this strong form of being almost 1-to-1 imply injectivity?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$. $\tilde{Y}$ is a ...
Veridian Dynamics's user avatar
3 votes
1 answer
121 views

Almost one-to-one endomorphism of minimal subshift?

Let $(X,T)$ be a minimal subshift. Can it happen that an endomorphism $\varphi\colon (X,T) \to (X,T)$ is almost 1-to-1 but not 1-to-1? Can it happen that a factor $\pi\colon (X,T) \to (Y,T)$ between ...
Veridian Dynamics's user avatar
11 votes
0 answers
212 views

Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)

Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...
Sophie M's user avatar
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2 answers
109 views

Minimal subshift with some $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?

There exists a minimal subshift $X$ with a point $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?
Veridian Dynamics's user avatar
6 votes
1 answer
222 views

Sliding block code on irreducible sofic shift

I was looking at the following exercise from Lind/Marcus book An Introduction to Symbolic Dynamics and Coding that I cannot solve. Can someone give me a hint? Find an example of a pair of irreducible ...
user19981007's user avatar
3 votes
1 answer
345 views

Entropy-minimal subshifts

Consider a subshift $X \subset \left\{0, \ldots, M \right\}^{\mathbb{N}}$. $X$ is said to be entropy-minimal if every subshift $Y \subsetneq X$ satisfies that $$h_{\mathrm{top}}(Y) < h_{\mathrm{top}...
Rafael Alcaraz Barrera's user avatar
1 vote
1 answer
88 views

Example of connected factor of symbolic system that is not a rotation

I am looking for an example of a factor $f\colon (X,T) \to (Y,T)$ between topological dynamical systems, where $(X,T)$ is a minimal subshift and $Y$ a connected topological space such that $(Y,T)$ is ...
Veridian Dynamics's user avatar
0 votes
0 answers
96 views

$||g_n||_{\infty} < \delta_{n-1}(g)$

It may be a simple question to post it here, but I posted this question in the Math Stack Exchange forum and no one answered me. Let $E$ be a (possibly infinite) alphabet and consider $X = E^{\...
Luísa Borsato's user avatar
8 votes
1 answer
280 views

Connection between entropy and the set of factors of a sequence

Let $a = (a_n)_{n=0}^\infty$ be a bounded real-valued sequence. By a factor of $a$ I mean a finite block $w \in \mathbb R^l$ that appears in $a$, that is, there exists $n \geq 0$ such that $a_n a_{n+1}...
Jakub Konieczny's user avatar
6 votes
3 answers
533 views

Ruelle-Perron-Frobenius theorem for shift of finite type

I know a version of Ruelle's theorem for expansive transformations in a compact metric space that says there is a single equilibrium state for a potential holder. In this Ruelle-Perron-Frobenius ...
Ilovemath's user avatar
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2 votes
0 answers
77 views

Equivalence between Gibbs measures and conformal measures

I was reading an article about Gibbs measures, but the author defines Gibbs measures in a different way than the usual (which is done by using conditional expectations). The way that he defines I have ...
Luísa Borsato's user avatar
6 votes
2 answers
319 views

Uniqueness of "Limit" of Cyclic Binary Strings

Set-up: By abuse, let $\sigma$ represent both the left shift operator on infinite bi-infinite strings and the cyclic left shift operator on finite strings. (Thus, for example, $\sigma(...01\bar{0}10......
Adam Quinn Jaffe's user avatar
3 votes
0 answers
72 views

Does an asymptotic component with large size in a minimal subshift always exist?

Let $(X, T)$ be a minimal subshift, i.e. $X$ is a closed $T$-invariant subset of $A^\mathbb{Z}$, where $T$ is the shift. A pair $x,y\in X$ is asymptotic if $d(T^nx, T^ny)$ goes to zero as $n\to\infty$....
Veridian Dynamics's user avatar
6 votes
0 answers
366 views

$C^{1+\epsilon}$ conjugacy of expanding map on circle

A continuously differentiable map $f:S^{1}\rightarrow S^{1}$ is called expanding if $|f^{'}(x)|>1$ for all $x\in S^{1}$. We can define the degree of f, def(f) to be number of preimage $f^{-1}(x)$, ...
Adam's user avatar
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7 votes
1 answer
245 views

Is the density of 1's in the Fibonacci word uniform?

The Fibonacci word is the limit of the sequence of words starting with $0$ and satisfying rules $0 \to 01, 1 \to 0$. Equivalently, it is obtained from the recursion $S_n= S_{n-1}S_{n-2}$ under ...
Darren Ong's user avatar
4 votes
0 answers
98 views

Weighted distribution of irrational rotation

Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
user119197's user avatar
3 votes
1 answer
273 views

On Krieger's Embedding Theorem

This is Theorem 10.1.1 of Lind & Marcus's book, An Introduction to Symbolic Dynamics and Coding. They say that is "straightfordward" to go from Let $X$ a shift of finite type and $Y$ a mixing ...
Veridian Dynamics's user avatar
6 votes
1 answer
361 views

continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only. Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
Michal's user avatar
  • 199
4 votes
1 answer
139 views

Asymptotic colouring of edges and vertices, and untwisting cocycles

This question regards colourings on edges and vertices on countable directed multigraphs. We start with an example. Let $G=\mathbb Z^2$. We define two functions $a_h$ and $a_v$ from $\mathbb Z^2$ to $\...
Alessandro Vignati's user avatar
3 votes
2 answers
416 views

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 $...
Claude's user avatar
  • 111
4 votes
1 answer
176 views

Measures maximizing entropy in a set of measures with fixed average for some observable

Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$. Consider $0<\alpha<1$ and let $$K_\alpha=\{...
Dominik Kwietniak's user avatar
4 votes
1 answer
132 views

The continuity of the the stable and unstable in definition of hyperbolic sets for flows

I would like to know whether the continuity of the stable and unstable subbundles $E^{s}$ and $E^{u}$ follows from the growth conditions as in the discrete case, or must be hypothesized, in the ...
Julian's user avatar
  • 41
42 votes
2 answers
3k views

Can one "hear" the shape of a polygon via external reflections?

This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?" A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-...
Joseph O'Rourke's user avatar
2 votes
1 answer
191 views

Synchronised $\beta$-shifts

I have been reading some papers recently, in particular, Blanchard's paper $\beta$-expansions and symbolic dynamics which state that a $\beta$-shift $S_{\beta}$ is a synchronised shift if and only if ...
Rafael Alcaraz Barrera's user avatar
3 votes
1 answer
392 views

Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not)

Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family. I wondered is it known that the MLC is not enough (or ...
yaoxiao's user avatar
  • 1,706
8 votes
1 answer
436 views

The graph of Rule 110 and vertices degree

Consider the elementary cellular automaton called Rule 110 (famous for being Turing complete): It induces a map $R: \mathbb{N} \to \mathbb{N}$ such that the binary representation of $R(n)$ is ...
Sebastien Palcoux's user avatar
3 votes
1 answer
125 views

Topological universality for Cantor maps

I am afraid this question might be very naïve, but I find it hard to locate a reference that does not answer a slightly different question. Consider the Cantor set $C$ and a continous map $f: C\to C$ ...
Benoît Kloeckner's user avatar
4 votes
0 answers
384 views

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 ${\rm C}^{\star}(X)$, ...
Sebastien Palcoux's user avatar
8 votes
0 answers
269 views

Shift on trivalent directed tree, operator and von Neumann algebra

Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...
Sebastien Palcoux's user avatar
2 votes
1 answer
238 views

Unique(ish) infinite string avoiding a set of patterns

Let $\Sigma$ be a finite alphabet of size at least 2. A (possibly infinite) string $s$ over alphabet $\Sigma$ encounters a pattern $p \in \mathbb{N}^*$ iff there is a non-erasing morphism $f: \mathbb{...
Mikhail Tikhomirov's user avatar
4 votes
1 answer
157 views

Subshifts with a free semigroup

Let $X$ be a subshift on a finite alphabet. I'm interested in the following property: there exist words $s,t\in\mathcal L(X)$ (the language of $X$) such that $\{s,t\}^*\subset \mathcal L(X)$. That is, ...
Nikita Sidorov's user avatar
1 vote
1 answer
120 views

Partitioning a subset of a subshift into comparable parts

Let $(X,\mu,\sigma)$ be a subshift on a finite alphabet, which we assume to be mixing. That is, for any cylinders $C, C'$ we have $\mu(\sigma^{-n}C\cap C')\to\mu(C)\mu(C')$ as $n\to+\infty$. We also ...
Nikita Sidorov's user avatar
4 votes
0 answers
177 views

Explicit symbolic codings

The short version of my question is that I need examples of explicit continuous symbolic codings of invertible dynamical systems. Here's a longer version. Suppose $(\Omega,\mu,T)$ is an invertible ...
Nikita Sidorov's user avatar