All Questions
Tagged with symbolic-dynamics ergodic-theory
57 questions
2
votes
1
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269
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Chaotic dynamics of maps on unit square that are NOT Triangular
We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
CommunityBot
- 1
11
votes
1
answer
521
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Cohomology for extension problems in symbolic/topological dynamics?
Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
2
votes
1
answer
183
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A sensitive 2-dimensional cellular automaton with a blocking word
I'am a Ph.D student in the domain of discrete dynamical systems. My thesis is about spectral properties of cellular automata in higher dimension.
Kurka gives a classification for one dimensional ...
- 171
3
votes
2
answers
432
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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 $\...
- 2,621
3
votes
1
answer
131
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the definition of the topological pressure for matrices
Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma_n$ the set of addmisible words with the ...
- 8,903
2
votes
1
answer
181
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Union of admissible words are subshift of finite type
Assume that $Q=(q_{ij})$ is a $k\times k$ with $q_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma_{Q}\rightarrow \Sigma_{Q}$, where
...
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1
vote
1
answer
237
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Proof that Sturmian shift is uniquely ergodic using irrational rotation
I am finding proofs of unique ergodicity of Sturmian shifts however I want to know if there is a proof that link that to the unique ergodicity of irrational rotations through conjugacy for example or ...
- 6,652
6
votes
0
answers
172
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Construction of minimal zero entropy measure-theoretically strong mixing subshift?
Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is
(1) minimal
(2) zero (topological) entropy
(3) measure-theoretically strong mixing (for some measure)?
I am in particular ...
- 2,621
0
votes
0
answers
54
views
Statistical characteristics of low complexity subshifts
I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
- 63.9k
2
votes
1
answer
205
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Exponential mixing for subshifts
I asked this question on Math.StackExchange some time ago and got no responses.
Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type
$$
\...
- 2,621
2
votes
1
answer
214
views
Irrational rotations are rank 2 by intervals without spacers
Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$.
S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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3
votes
1
answer
171
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Does full shift have the local product structure?
We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is ...
- 1,043
1
vote
0
answers
212
views
Is there a condition for a subshift of finite type to be uniquely ergodic?
Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic?
- 63.9k
1
vote
0
answers
61
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Number of permitted words up to permutation in a subshift
Let $A$ be a finite set and let $X \subseteq A^{\mathbb{N}}$ be a subshift. Let $\mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w \in \mathcal{L}_n$, one can ...
- 323
1
vote
1
answer
107
views
Lyapunov spectrum($h_{\mathrm{top}}(K(\alpha))$) achieves a positive value somewhere
$\DeclareMathOperator{\top}{\mathrm{top}}$Let $(\Sigma, T)$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Hölder continuous map.
Let
$$K(\alpha)=\Big\{x\in \...
- 1,043
6
votes
1
answer
266
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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 ...
- 8,903
6
votes
2
answers
238
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Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing
The following result is on page 26 of this paper by Ferenczi [PDF].
Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) \...
- 23.2k
11
votes
0
answers
212
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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: ...
- 63.9k
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 ...
- 8,903
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 $f^{-1}(x)$, ...
- 1,043
4
votes
0
answers
98
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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, ...
- 675
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=\{...
- 8,903
2
votes
1
answer
421
views
Measure of large cylinder sets
Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an $\epsilon$-portion, all cylinder sets of length $n$ (large enough) have $m$-measure between $exp(-nh-...
- 1,228
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 ...
- 18.7k
7
votes
3
answers
217
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 $...
- 330
15
votes
0
answers
3k
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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 ...
- 1,658
6
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0
answers
255
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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 ...
CommunityBot
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3
votes
2
answers
174
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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}$ ...
- 330
0
votes
2
answers
809
views
Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure?
I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:
Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset \...
- 307
3
votes
0
answers
209
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$...
- 2,560
3
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2
answers
340
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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\}^{\...
- 1,658
1
vote
0
answers
139
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-* ...
- 105k
4
votes
1
answer
273
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What is known about first return times to Markov partitions for Anosov diffeomorphisms?
Consider an Anosov diffeomorphism $T: M \rightarrow M$ and a corresponding Markov partition $\mathcal{R}$ of $M$. For $x \in M$, let $\mathcal{R}(x)$ denote the element of $\mathcal{R}$ containing $x$ ...
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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 $...
- 63.9k
0
votes
1
answer
172
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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 ...
CommunityBot
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2
votes
1
answer
267
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 $\bar{d}$ ...
CommunityBot
- 1
4
votes
2
answers
445
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}\...
- 2,459
5
votes
1
answer
543
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 ...
- 2,244
1
vote
1
answer
410
views
joining or coupling
given two shift invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$, is there a way to construct joinings of them? It's very diffcult, in general, to find exactly the minimal joining i.e, ...
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 ...
- 18.7k
1
vote
1
answer
176
views
Find a sequence with uniform frequencies and recurrent property
Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$
such that this sequence has uniform ...
CommunityBot
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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.
...
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5
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0
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142
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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 ...
- 3,597
2
votes
2
answers
209
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Mixing coded systems and period of their graph presentations
A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8.
(http://www.sciencedirect.com/...
7
votes
3
answers
792
views
A weak-mixing, zero entropy measure on the 2-shift which gives equal weight to both symbols
I am currently sketching a paper in the general area of symbolic dynamics in which I would like to be able to use the following fact:
Proposition (proposed): there exists a shift-invariant ...
CommunityBot
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0
votes
2
answers
305
views
The Book for ergodic theory on SFT in dimension $D>1.$
I have been unable to find a good reference for a book that study in details ergodic theory on sub shifts of finite type in dimension $D>1.$ The only reference that I got was actually a book by ...
7
votes
1
answer
292
views
Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture
Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
- 2,758
2
votes
1
answer
147
views
Does conjugacy preserve the set of synchronizing blocks?
A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then $uvw$...
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3
votes
1
answer
206
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Automorphisms of strictly ergodic shift spaces
Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I ...
- 170
8
votes
1
answer
2k
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Intuition of Kolmogorov-Sinai entropy
For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself.
For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define
$h(\mathcal{A},...
- 2,758