All Questions
25 questions
3
votes
1
answer
90
views
Asymptotic growth rate for primitve S-adic systems
It is known that for a primitive substitution $S:\mathcal{A}\to \mathcal{A}^+$, there exists constants $c,C>0$ such that
$$ c\theta_S^n \leq \vert S^n(a)\vert \leq C \theta_S^n \quad \text{for all} ...
- 633
6
votes
0
answers
94
views
Automorphism groups of subshifts and factor maps
Let $\pi : X \to Y$ be a factor map between subshifts over finite alphabets.
Let $\operatorname{Aut}(X)$ and $\operatorname{Aut}(Y)$ stand for automorphism groups of these shifts.
We say that $\varphi ...
- 1,658
1
vote
1
answer
148
views
Approximation of subshifts in Hausdorff distance
I have recently been interested in some questions which stem from taking subshifts which converge to a limiting subshift in the Hausdorff metric.
More specifically, given an alphabet $\mathcal{A}$, I ...
- 633
0
votes
0
answers
120
views
Growing gliders under rule 110
I found a glider in the evolution space of rule 110 that grows constantly in size. Normal gliders live in the so-called ether, e.g. the so-called E-glider:
Other – often complex – gliders exist in an ...
- 9,720
2
votes
1
answer
129
views
Reference on relation between SFTs and Wang-tiles
I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a ...
- 633
4
votes
2
answers
273
views
Lower bounds for pattern complexity of aperiodic subshifts
In the setting of symbolic dynamics over $\mathbb{Z}^d$, one can define for the $n$-th pattern complexity of a given a subshift $\Omega\subseteq \mathcal{A}^{\mathbb{Z}^d}$ as
$$ c_n(\Omega):= \Big\...
- 633
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 ...
- 17k
1
vote
0
answers
55
views
Continuity of Kneading invariants of generalised $\beta$-trasformations
For $\beta \in (1,2]$ and $\alpha \in [0,2-\beta]$ consider the generalised $\beta$-transformation $T_{\alpha,\beta}:[0,1] \to [0,1]$ to be $$T_{\alpha, \beta}(x) = \beta x + \alpha \mod 1.$$ It is a ...
6
votes
1
answer
266
views
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 ...
- 1,043
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}...
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
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 ...
- 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 ...
- 1,658
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 ...
- 1,658
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 $\bar{f}$ ...
- 1,658
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 $x_k, |k|>n$.
...
- 2,135
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?
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 ...
4
votes
1
answer
224
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 ...
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
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 $uvw$...
- 1,658
2
votes
2
answers
209
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.
(http://www.sciencedirect.com/...
- 1,658
2
votes
1
answer
344
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
3
answers
772
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 ...
- 701
6
votes
2
answers
1k
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 ...
- 18.7k