# Tagged Questions

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### 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 $. ...

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86 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 ...

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83 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 ...

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48 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) ...

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227 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 ...

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136 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 ...

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**1**answer

80 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- ...

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124 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 ...

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194 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 ...

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116 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 ...

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144 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.
...

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126 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 ...

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**1**answer

116 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 ...

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328 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
...

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132 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 ...

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250 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 ...

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282 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 $\pi(u)=y$ ...

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395 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 ...

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224 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 ...

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256 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 ...

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**1**answer

272 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 ...

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187 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 ...

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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 ...

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335 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 ...

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529 views

### Periodic sequences in symbolic dynamics

I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of ...

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647 views

### topologically mixing subshifts without ergodic measures

Are there examples of subshifts (that is, closed shift-invariant subsets of the full shift {$1...n$}${}^{\mathbb{Z}}$) on which the shift is topologically mixing, which admit a shift-invariant ...

**3**

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181 views

### 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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1k views

### How can generic closed geodesics on surfaces of negative curvature be constructed?

As far as I understand it the closing lemma implies that closed geodesics on surfaces of negative curvature are dense. So: how can they be constructed in general?
A concrete answer that dovetails ...

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465 views

### A regularity property of transition matrices for the cat map

I've noticed a rather strange phenomenon (not important for my particular research, but interesting) and wouldn't be surprised if someone more versed in symbolic dynamics (i.e., just about anyone who ...

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1k views

### Proper families for Anosov flows

So I've been skimming Bowen's 1972 paper "Symbolic Dynamics for Hyperbolic Flows" hoping it would give me some insight into how to build a Markov family for the cat flow (i.e., the Anosov flow ...