All Questions
Tagged with ds.dynamical-systems symbolic-dynamics
129 questions
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 ...
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 ...
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: ...
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 $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?
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 ...
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}...
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 ...
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^{\...
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}...
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 ...
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 ...
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$....
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)$, ...
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, ...
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 ...
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 ...
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 $\...
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 $...
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=\{...
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 ...
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 ...
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 $f: C\to C$ ...
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, ...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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 $\varphi: [0,1]\to [0,1]$, $...
6
votes
0
answers
341
views
Had this theorem in Tresser's article been proven somewhere?
The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
15
votes
0
answers
3k
views
Weak$^*$ 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 ...
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 ...
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 $\bar{f}$ ...
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 $x_k, |k|>n$.
...
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\}^{\...
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 ...
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 $...
4
votes
2
answers
444
views
Entropy equals zero?
Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = \mathcal{P}\...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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.
...
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 ...
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 $. ...
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 ...
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 $\...