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3 votes
2 answers
137 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 ...
Veridian Dynamics's user avatar
3 votes
1 answer
120 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
  • 695
4 votes
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
343 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
279 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
531 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
  • 677
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
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
  • 1,043
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
360 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
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
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
119 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
3 votes
2 answers
426 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 ...
Nikita Sidorov's user avatar
9 votes
1 answer
210 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 ...
Christian Remling's user avatar
5 votes
1 answer
289 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\...
Nikita Sidorov's user avatar
15 votes
2 answers
648 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 ...
Vim's user avatar
  • 253
0 votes
1 answer
231 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 $\varphi: [0,1]\to [0,1]$, $...
aglearner's user avatar
  • 14.3k
6 votes
0 answers
341 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 ...
Evgeny's user avatar
  • 165
15 votes
0 answers
3k 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 ...
Dominik Kwietniak's user avatar
6 votes
0 answers
255 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 ...
Dominik Kwietniak's user avatar
3 votes
2 answers
174 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 $\bar{f}$ ...
Dominik Kwietniak's user avatar
9 votes
0 answers
601 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 $x_k, |k|>n$. ...
Nikita Sidorov's user avatar
3 votes
2 answers
340 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\}^{\...
Bruno Brogni Uggioni's user avatar
0 votes
0 answers
143 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 ...
Alan's user avatar
  • 1,594
0 votes
0 answers
182 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 $...
Bruno Brogni Uggioni's user avatar
4 votes
2 answers
444 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}\...
Bruno Brogni Uggioni's user avatar
2 votes
1 answer
323 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?
Jesse Solomon Scott's user avatar
5 votes
1 answer
542 views

When is the time one map of a suspension flow ergodic?

I'm sure the answer to the following question is well known but I couldn't find the answer I needed. Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure ...
Tom Kempton's user avatar
3 votes
2 answers
398 views

Does the 2-shift map have a root automorphism?

By the 2-shift map I mean the map $T:\{0,1\}^\mathbb{Z}\to \{0,1\}^\mathbb{Z}$ that shifts the sequence leftwise. By a root I mean an homeomorphism $\psi:\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ that ...
EvaristoCarriego's user avatar
2 votes
1 answer
261 views

Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...
Rafael Alcaraz Barrera's user avatar
13 votes
2 answers
775 views

On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e., $$ \mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}. $$ Then, as is well known, $\mathcal T$ has a non-empty ...
Nikita Sidorov's user avatar
14 votes
2 answers
2k views

Perron-Frobenius theory for reducible matrices

Can someone suggest some sources/references dealing with the Perron-Frobenius theory for nonnegative matrices that are reducible? Specifically, if $A\ge 0$ is a $d\times d$ matrix with no assumptions ...
Ilya Kapovich's user avatar
3 votes
0 answers
195 views

Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't. ...
Tom Kempton's user avatar
6 votes
3 answers
319 views

Approximating Subshifts From Below

I'm looking to understand how to approximate certain countable alphabet subshifts by Markov shifts, and realised that I don't know how to do it even in the finite alphabet case. My guess is that the ...
Tom Kempton's user avatar
1 vote
0 answers
111 views

The value of the sequence generated by the substitution

Given a substitution $1\to 100$, $0\to 01$, then we have $1\to 100\to1000101\to10001010110001100\to\cdots$, we denote this limits (fixed point of this substitution) as $(a_n)$, given $\beta>1 $. ...
Ben Ben's user avatar
  • 115
5 votes
0 answers
142 views

Growth in families of trees

I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome. Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set ...
Tom Kempton's user avatar
3 votes
2 answers
431 views

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 $\...
Ben Ben's user avatar
  • 115