# 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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### 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 ...

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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 $x$...

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80 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
$$
\phi: [0,1]\times D_n\rightarrow D_n,
$$
...

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138 views

### 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}} ...

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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 ...

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221 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 $...

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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 ...

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### 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 ...

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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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120 views

### 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 ...

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83 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 ...

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86 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: ...

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81 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 $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?

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### 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 ...

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140 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}...

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### 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 ...

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82 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^{\...

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226 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}...

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237 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 ...

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64 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 ...

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254 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......

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### 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 $n\to\infty$....

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187 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 $f^{-1}(x)$, ...

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163 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 ...

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84 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, ...

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136 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 ...

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269 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 ...

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### 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 $\...

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246 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 $...

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### 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=\{...

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### 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 ...

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### 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-...

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### 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 ...

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### 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 ...

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325 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 ...

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99 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 $f: C\to C$ ...

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204 views

### Extension of Coburn's theorem on isometry and Toeplitz algebra

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 ${\rm C}^{\star}(X)$, the ${\rm C}^{\star}$-algebra ...

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### 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 $\...

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180 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{...

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### 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, ...

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### 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 ...

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170 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 ...

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165 views

### Besicowitch distance between sets of invariant measures, ergodic vs non-ergodic

When working with Dominik Kwietniak and Jakub Konieczny, the question appeared:
Let $X$ and $Y$ be two subshifts on the same alphabet, $M(X)$, $M(Y)$ the sets of shift-invariant measures on $X$ and $...

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250 views

### Subshifts of finite type of guaranteed positive entropy

Let $\Sigma$ be a subshift of finite type (SFT) with the alphabet $\{0,1\}$, which is given by the set of forbidden words $\mathcal F$, all of length $N$.
Question. Is there a $\delta>0$ such ...

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### Can the full shift be embedded in a flow?

Write $I=[0,1]$, and let $S$ be the shift on $X=\{ (x_n)_{n\in\mathbb Z} : x_n\in I^k \}$. Is there a flow $\phi_t$ on $X$ with $\phi_1=S$? Here I require that $\phi_t$, for fixed $t$, is at least a ...

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529 views

### Sequences with 3 letters

For a positive integer $n$ I would like to construct long sequences consisting of 0, 1 and 2's such that for any two subsequences consisting of $n$ consecutive elements the number of 0's , 1's or 2'...

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226 views

### The spectral radius of a binary matrix - polynomial growth?

(This is a follow-up to The spectral radius of a binary matrix)
Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$.
QUESTION. Is there a $\delta\in\bigl(0,\frac12\...

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610 views

### Is the following series consisting of equally distributed $\pm 1$ bounded?

Apologise in advance if this problem isn't research-level (I'm quite certain it isn't). It's just I found it quite intriguing because it turned out to be much more subtle than it appeared at my first ...

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### Beginners level question : symbolic dynamics and notations

Let $f(.)$ be a chaotic 1 D Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control ...

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### Inverse map of chaotic map : confusion and request for information

This is based from the paper titled, "Chaos-Based Simultaneous Compression and Encryption for Hadoop" in Section 2.3.1 download link
The Authors say that given a symbolic sequence, it can be encoded ...