**3**

votes

**2**answers

163 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} = ...

**2**

votes

**1**answer

88 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?

**0**

votes

**1**answer

85 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, ...

**4**

votes

**1**answer

95 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 ...

**7**

votes

**0**answers

177 views

### Nice sign-expansions of special surreal numbers

What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic?
I can think of more than one natural way to ...

**1**

vote

**1**answer

70 views

### Terminology for set of infinite strings with a certain prefix

Let $\mathcal{A}$ be a finite alphabet, and let $C$ be the Cantor space $\mathcal{A}^\omega$ under the product topology.
Given a finite string $s \in \mathcal{A}^*$, let $C(s)$ be the set of all ...

**3**

votes

**2**answers

149 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

191 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 ...

**1**

vote

**0**answers

77 views

### Periodic configurations for elementary cellular automata

Let $L$ be an elementary cellular automaton. Then $L$ acts on $\{0,1\}^{\mathbb{Z}}$. We say that a configuration $w\in\{0,1\}^{\mathbb{Z}}$ is periodic if $L^{(n)}(w)=w$ for some $n\in\mathbb{N}$.
...

**11**

votes

**2**answers

489 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 ...

**9**

votes

**1**answer

310 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

85 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.
...

**4**

votes

**3**answers

198 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

93 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

115 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

**1**answer

136 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 ...

**5**

votes

**1**answer

96 views

### 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) ...

**2**

votes

**2**answers

259 views

### Brun's algorithm

Does anyone have an exact reference for the weak convergence (convergence in angle) of Brun's subtractive multi-dimensional continued fractions algorithm (in all dimensions)? I have been given ...

**7**

votes

**1**answer

239 views

### Über theorem on unavoidable patterns?

Let $A$ be an alphabet of $k$ symbols,
and $p$ a pattern.
An example of a pattern is $p=XX$, where $X$ is any finite
string of symbols from $A^+$.
Avoiding $p$ is avoiding any subword repeated twice ...

**1**

vote

**1**answer

215 views

### A problem in symbolic dynamics

I got a fun problem.
Define the alphabet $\mathcal{A}=\{0,1,2\}$ and the set $\mathcal{A}^{\leq n}=\{ x_1x_2\ldots x_n: x_i\in \mathcal{A}\}$ of words of length $n,$ for each $n\in\mathbb{N}.$
...

**3**

votes

**1**answer

71 views

### For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) sup_{x \in [i]}f(x) )$ is bounded

Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that
$$|f|_{var}= \sum_{n=1}^{\infty} var_n f < \infty,$$
where $var_n f = sup\{|f(x)-f(y)|: x,y \in X , x_k = y_k, ...

**1**

vote

**1**answer

150 views

### Cycles in directed graphs

Let G be a finite directed graph (allowing multiple edges). We define a cycle (as usual) to be a sequence of edges $e_0, e_1, \dots, e_{n-1}$ (up to cyclic permutation) such that the terminal vertex ...

**1**

vote

**1**answer

107 views

### Group action of $G<\mathbb Z^\infty_2$ over the Golden mean shif [closed]

I'm am looking for an action of an infinite subgroup of $\mathbb Z^\infty_2$ over the golden mean shift space $$X=\{x\in \{0,1\}^\mathbb N : x_i=1\Rightarrow x_{i+1}=0\}$$ such that any element of $G$ ...

**0**

votes

**1**answer

148 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 ...

**5**

votes

**0**answers

249 views

### Deterministic shifts

We consider (topological) dynamical systems $(\Omega, S)$, where $S$ is the shift $(Sx)_n=x_{n+1}$, and $\Omega\subset[0,1]^{\mathbb Z}$ is a compact, shift invariant subspace. I call such a system ...

**4**

votes

**1**answer

152 views

### Graph presentation of Lexicographic shifts

Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...

**2**

votes

**1**answer

85 views

### Constructing an interval exchange given a prescribed trajectory

Given a prescribed trajectory, is it possible to construct an interval exchange having this trajectory?
For example, given a 3-letter word (like aaabbbccabcaaa ), is it possible to construct a 3- ...

**1**

vote

**1**answer

165 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 ...

**0**

votes

**2**answers

177 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 ...

**6**

votes

**1**answer

219 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 ...

**6**

votes

**3**answers

452 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 ...

**2**

votes

**1**answer

122 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 ...

**2**

votes

**2**answers

168 views

### 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.
...

**3**

votes

**1**answer

147 views

### 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 ...

**3**

votes

**1**answer

171 views

### Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by ...

**4**

votes

**1**answer

490 views

### 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
...

**1**

vote

**2**answers

147 views

### Estimation of number of ways to concatenate strings of the form $01^k2^k$ to create a string of length n

In symbolic dynamics, the context-free shift is the set of biinfinite concatenations of strings of the form $01^k2^k$ for $k\in\mathbb{N}\cup\lbrace 0\rbrace$. I've reduced a certain problem to ...

**28**

votes

**5**answers

2k views

### What is the effect of adding 1/2 to a continued fraction?

Is there a simple answer to the question "what happens to the continued fraction expansion of an irrational number when you add 1/2?" A closely related question is "what happens to such an expansion ...

**2**

votes

**2**answers

255 views

### probability measures with entropy equal to nonnegative number

Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this ...

**6**

votes

**1**answer

294 views

### Relative irreducibility

Let $X$ be a one-dimensional one-step irreducible shift of finite type and let $\pi$ be a one-block factor code from $X$ to a sofic $Y$. Suppose $y$ is a right transitive point of
$Y$ and $\pi(u)=y$ ...

**6**

votes

**1**answer

419 views

### Aproximating dynamical systems by intrinsically ergodic systems

Let $X$ be a compact metric space and $f:X \to X$ a continuous map. We say that $(X,f)$ is approximated from below by a sequence of compact metric spaces $(X_i)_{i \geq 1}$ and a sequence of ...

**1**

vote

**0**answers

154 views

### Entropy of factors of Bernoulli schemes

Let $X$ be a Bernoulli scheme. A factor $\psi: X \to Y$ is finitary if for almost every $x \in X$ there exist integers $m \leq n$ such that the zero coordinates of $\psi(x)$ and $\psi(x')$ agree for ...

**0**

votes

**1**answer

306 views

### Recurrence theorems

Bonjour, I need some help regarding recurrence theorems in shift spaces. I am aware of Poincaré's Recurrence Theorem, but I'm sure I've heard of another one telling about the time required to get back ...

**2**

votes

**1**answer

251 views

### Coded Systems and dense subsets

A shift space $(X, \sigma)$ is a coded system if there exist a countable collection of finite words $(\omega^n)_{n \in \mathbb{N}}$, called generators, such that $X$ is the closure of the set of ...

**5**

votes

**1**answer

267 views

### Limits of intrinsically ergodic systems

Let $(X_i)$ be a sequence of compact metric spaces and $(f_i)$ a sequence of transitive transformations $f_i:X_i \to X_i$ with $0 < h_{top}(f_i) < \infty$.
The sequence of dynamical systems ...

**0**

votes

**1**answer

276 views

### Modulo dynamics on [0,1)

For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of ...

**2**

votes

**1**answer

289 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 ...

**2**

votes

**0**answers

198 views

### Does an aperiodic dynamical system have $n$-markers for any $n$?

I was wondering if a certain lemma in an article by Downarowicz holds in a more general setting (see details below):
Let $(X,T)$ be a topological dynamical system. I.e. $X$ is a compact Hausdorff ...

**1**

vote

**0**answers

95 views

### Finitary factors of Bernoulli schemes that pair duals

This question is related to my question:
entropy preserving finitary factor maps of Bernoulli schemes.
Hopefully, this one is a bit easier.
Let $X=\{0,1\}^\mathbb{Z}$ with measure ...

**3**

votes

**2**answers

336 views

### How to detect frequency?

Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or
closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is
irrational. Consider in $\mathbb{S}^{1}$ the sequence ...