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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8
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1answer
165 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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3answers
156 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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35 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 ...
6
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2answers
171 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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0answers
36 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 $n\to\infty$....
5
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135 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 $f^{-1}(x)$, ...
7
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1answer
146 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 ...
4
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73 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, ...
3
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1answer
106 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 ...
2
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1answer
224 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 ...
4
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1answer
121 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 $\...
2
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1answer
183 views

fast algorithms for external angle computations

Two related problems related to the complex quadratic polynomial $f_c(z) = z^2 + c$ and Mandlebrot and/or Julia sets: find an external angle $\theta_c$ for a complex point $c$ find a complex point $...
3
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1answer
114 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=\{...
4
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1answer
80 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 ...
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2answers
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-...
2
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1answer
99 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 ...
3
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1answer
102 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 ...
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234 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 ...
3
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1answer
94 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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184 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 ...
7
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165 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 $\...
2
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1answer
167 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{...
4
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1answer
118 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, ...
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1answer
108 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 ...
4
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0answers
167 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 ...
7
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3answers
161 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 $...
3
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2answers
215 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 ...
9
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1answer
154 views

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 ...
5
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1answer
197 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\...
14
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2answers
592 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 ...
1
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1answer
197 views

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 ...
1
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1answer
173 views

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 ...
0
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1answer
132 views

A modified Cantor and its measure

Recall that Cantor set can be defined as the set of numbers in $[0,1]$ that don't contain $1$ when written in ternary number system. Alternatively if we consider the map $\varphi: [0,1]\to [0,1]$, $...
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0answers
320 views

Had this theorem in Tresser's article been proven somewhere?

The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
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0answers
820 views

Weak$^*$ convergence of measures vs. convergence of supports

Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
5
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0answers
150 views

Completeness of the space of measures under $d$-bar metric

Does anybody know the reference to a proof of the following fact (which is not hard to prove, but seems to be well-known, see here): The space of shift-invariant measures under Ornstein's d-bar metric ...
6
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1answer
200 views

What is the simplest SFT on $\mathbb{Z}^2$ that has no periodic points?

An SFT (shift of finite type) is a set of maps to some finite alphabet that is defined by a finite number of disallowed finite words. By simple I mean has a small alphabet and a small number of ...
3
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2answers
131 views

a bound for Feldman's **f-bar** $\bar{f}$ metric for measures

My question regards properties of the f-bar metric $\bar{f}$ defined for shift invariant measures on $\mathscr{A}^\infty$ where $\mathscr{A}$ is a finite alphabet. The definition of the $\bar{f}$ ...
7
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1answer
152 views

When do automorphisms of subshifts extend to automorphisms of the full shift?

Let $A$ be a finite alphabet, $X$ = $(A^\mathbb{Z}, \sigma)$ the full shift, and $Y \subset X$ a subshift. Question: Are there any general results characterizing whether automorphisms of $(Y, ...
2
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1answer
128 views

Reference for one-sided subshifts

A well known result in Symbolic Dynamics asserts that every two-sided subshift on a finite alphabet necessarily consists of all doubly infinite words not containing any finite word from a given set of ...
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459 views

Topological entropy and periodic sequences of a subshift

Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$. ...
3
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0answers
192 views

Repartition of 1's in the “Chacon word”

Consider the "Chacon words": $B_0=0$ and $B_{n+1} = B_nB_n1B_n$. The word $B_n$ has $\ell_n := \frac{3^{n+1}-1}{2}$ digits and the number of $1$'s in $B_n$ is $\ell_n - 3^n = \ell_{n-1} \sim \ell_n/3$...
3
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2answers
207 views

Convex combinations of Bernoulli Measures

How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures? I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\...
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0answers
122 views

weak-* versus entropy growth

General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...
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0answers
119 views

A question from One Dimensional Dynamics book by De-Melo and van-Strien

In One Dimensional Dynamics, on page 27 I don't understand how does $(1.7)$ follow; anyone care to explain this to me? Thanks in advance. I am adding some information from the text below: We are ...
0
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0answers
174 views

On a certain set of probability measures on a shift

Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2. Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where $...
0
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1answer
151 views

entropy growth of invariant measures - General question

In general, given a sequence of shift-invariant measures $\eta_{n}$ on $\{0,1\}^{\mathbb{N}}$ what to do to guarantee this convergence of entropies: $$h(\eta_{n}) \rightarrow \log2?$$ Because I'm ...
2
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1answer
197 views

entropy and d-bar: how do we estimate continuity?

Let $G = \{0,1\}^{\mathbb{N}} = \mathbb{Z}_{2}^{\mathbb{N}}$ be the Bernoulli space of two symbols, let $\sigma$ be the shift map and $M(G)$ the set of $\sigma$-invariant probabilities. Let $\bar{d}$ ...
4
votes
2answers
367 views

Entropy equals zero?

Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$ in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define $\mathcal{P} = \{[0],[1]\}$; $\mathcal{P}^{n} = \mathcal{P}\...
2
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1answer
197 views

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?