34
votes
Can one "hear" the shape of a polygon via external reflections?
For question 3, the answer is yes: take a solid disc and excavate half of the Penrose unilluminable room from it. Then, there are boundary arcs which can never be touched, and you can perturb them ...
- 12.4k
17
votes
Accepted
Is the following series consisting of equally distributed $\pm 1$ bounded?
The sequence $\sum a_n$ is unbounded.
This is a consequence of a general result from Kesten,
On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The ...
- 18.7k
16
votes
Open problems in symbolic dynamics
Mike Boyle once compiled a pretty large collection of open problems in symbolic dynamics, and has been keeping track of their status.
11
votes
Accepted
What is the simplest SFT on $\mathbb{Z}^2$ that has no periodic points?
Wang tiles are unit squares with edges marked with colors, and the problem of whether a given set of Wang tiles can tile the plane such that edges of adjacent squares match has been studied ...
- 2,758
10
votes
Accepted
Is the density of 1's in the Fibonacci word uniform?
Yes. By Proposition 2.1.10 in Lothaire, Algebraic Combinatorics on Words, if $u$ is any substring of the Fibonacci word then
$$\left| \frac{\mbox{number of $1$'s in $u$}}{\mbox{length of $u$}} - \...
- 156k
9
votes
Can the full shift be embedded in a flow?
For odd $k$, the shift in $X^k$ has no square root (and hence does not lie in a 1-parameter subgroup). Indeed, the the set of 2-periodic points can be identified with $(I^k)^2$, where the shift acts ...
- 63.9k
9
votes
Accepted
Ruelle-Perron-Frobenius theorem for shift of finite type
The most intuitive explanation I know is the following: suppose that you have a certain amount of mass (I usually picture a pile of sand) that is distributed over $\Sigma_A^+$ according to the density ...
- 8,903
8
votes
Is the following series consisting of equally distributed $\pm 1$ bounded?
I've decided to upgrade my comments and make an answer out of them, even though I'm just addressing the (easier) variant suggested by the OP at the end of the post, where we replace the golden ratio ...
- 24.2k
8
votes
Accepted
When do automorphisms of subshifts extend to automorphisms of the full shift?
If $\phi$ is an automorphism of $X$ and $Y$ is the set of points in $X$ of exact period $n$, then $\phi|_Y$ is an automorphism of $Y$. There is a subtle relationship between the sign of the ...
- 2,758
8
votes
Accepted
Subshifts with special property
Define $X(m)$ as the image of $X$ in $(A^m)^\mathbf{Z}$, mapping $(a_n)_{n\in\mathbf{Z}}$ to $((a_{n+k})_{0\le k<m})_{n\in\mathbf{Z}}$. This is an equivariant embedding.
Fix $N$. We claim that if $...
- 63.9k
8
votes
Accepted
Topological dynamical systems with only zero-entropy factors
This question is very related to the question of lowering topological entropy, introduced in ``Can one always lower topological entropy?'' by Shub and Weiss and then very nearly solved by ...
- 2,621
7
votes
Open problems in symbolic dynamics
You mentioned substitution systems, so the Pisot substitution conjecture obviously has to be mentioned. There's a (mostly) up to date exposition by Akiyama, Barge, Berthé, Lee and Siegel that can be ...
7
votes
Lower bounds for pattern complexity of aperiodic subshifts
There are a few things to clarify here.
First of all, the two-dimensional version of Morse-Hedlund, i.e. that whenever $X$ contains a point with no period vector, $p_{m,n}(X) \geq mn+1$ holds for all $...
- 2,621
6
votes
Accepted
Measures maximizing entropy in a set of measures with fixed average for some observable
In the setting you describe, for each $\alpha \in (0,1)$ the $(1-\alpha,\alpha)$-Bernoulli measure is the unique measure achieving the maximum. The function $\alpha \mapsto \eta(\alpha)$ is the ...
- 8,903
6
votes
Accepted
Entropy-minimal subshifts
Let $f$ be a sublinear function that tends to infinity, such as $f(n) = \sqrt{n}$. Define $X \subset \{0,1,2\}^{\mathbb{N}}$ by forbidding all long enough words $w$ with more than $f(|w|)$ occurrences ...
- 740
6
votes
Accepted
A unique equilibrium state which does not have Gibbs property
The measure $\mu$ does not necessarily have the Gibbs property. In fact, it has the Gibbs property if and only if $f$ has the Bowen property: $\sup_n \sup \{ |S_n f(x) - S_n f(y)| : x_1 \dots x_n = ...
- 8,903
6
votes
Accepted
Word combinatorics terminology question
Yes you find these in all infinite mixing SFTs. More is true. As mentioned, these words are sometimes called unbordered, I'll use that word.
The following is Theorem 8.3.9 in [1].
Theorem. Let $x \in ...
- 6,652
5
votes
Accepted
Topological universality for Cantor maps
No.
Your condition is called being a (topological) subshift. If $(C,f)$ is a topological subshift, then there exists a finite clopen partition $P$ of $C$ such that the family $(f^{-n}P)_{n\ge 0}$ ...
- 63.9k
5
votes
Accepted
Decidability of periodic tilings of the plane
Deciding whether a set of tiles admits a periodic tiling or no tiling at all is undecidable as well.
This has been shown in Y.S. Gurevich, I.I. Koryakov, Remarks on Berger's paper on the domino ...
- 244
5
votes
Accepted
Sequences with 3 letters
Here is a sketch of a proof that there are no such complete sequences for $n>4$.
Consider the graph where the vertices are the triples of nonnegative integers that sum to $n$ and construct
an edge ...
- 376
5
votes
Accepted
Connection between entropy and the set of factors of a sequence
Here's an attempt. Let me restrict to functions with values in $[0,1]$ and my entropies are computed with binary log.
If we consider $X \subset [0,1]^{\mathbb{Z}}$ with the compact topology obtained ...
- 6,652
5
votes
Accepted
Does this strong form of being almost 1-to-1 imply injectivity?
No. Consider an irrational rotation $R$ of the circle (which I identify with [0,1)) by an angle $\alpha$. Let $\alpha<\beta<1$ be a point not lying in the orbit of 0 under $R$. Set $A_1=[0,\...
- 23.2k
4
votes
Accepted
A modified Cantor and its measure
This is called a cookie cutter. If by smooth, you mean $f$ is $C^{1+\epsilon}$ or smoother, then it's known that $f$ preserves a fully supported absolutely continuous invariant measure on $[0,1)$. In ...
- 23.2k
4
votes
Accepted
The spectral radius of a binary matrix - polynomial growth?
A paper that seems to directly address your question is the 1987 paper of Brualdi and Solheid, *On the minimal spectral radius of matrices of zeros and ones". That paper shows that if the number of 1'...
- 23.2k
4
votes
The graph of Rule 110 and vertices degree
I guess I'll indulge in my guilty pleasure a bit. The connected component of the number $1$ has unbounded degree.
Let $X = \{x \in \{0,1\}^{\omega} \;|\; \sum x < \infty\}$, the finite support ...
- 6,652
4
votes
Accepted
Subshifts with a free semigroup
For an irreducible sofic shift which is not periodic you will have this property. The Fischer cover gives a strongly connected deterministic partial automaton with all states initial and final ...
- 38.6k
4
votes
Ruelle-Perron-Frobenius theorem for shift of finite type
I am not sure if this is really what you are asking for but here is a finite dimensional version of some results of the Perron-Frobenius theorem. I hope this helps you better understand what happens.
...
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4
votes
Sliding block code on irreducible sofic shift
This is a fun pair of exercises (the first one you mention is 3.2.9 and the second is 2.3.6a)! For 2.3.6a, recode to a $1$-block code $\phi$ on an irreducible edge shift $X$, suppose that $x, x' \in X$...
- 695
4
votes
Accepted
Minimal subshift with some $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?
We can produce such a subshift by a standard hierarchical construction.
Let $w_{0,0} = 01$ and $w_{0,1} = 011$.
For each $k \geq 0$, define $w_{k+1,0} = w_{k,0} w_{k,0} w_{k,1}$ and $w_{k+1,1} = w_{k,...
- 740
4
votes
Accepted
Does full shift have the local product structure?
No. Take $\mu$ to be a measure supported on a Sturmian shift corresponding to some irrational rotation $R_\alpha$. If $\mathcal F^+$ denotes the $\sigma$-algebra generated by the coordinates in $\...
- 23.2k
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