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 ...
Adam P. Goucher's user avatar
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 ...
coudy's user avatar
  • 18.5k
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 ...
Douglas Lind's user avatar
  • 2,748
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 $u$}} - \...
David E Speyer's user avatar
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 ...
Vaughn Climenhaga's user avatar
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 ...
Christian Remling's user avatar
8 votes
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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 ...
Douglas Lind's user avatar
  • 2,748
8 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 ...
YCor's user avatar
  • 60.1k
8 votes
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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 $...
YCor's user avatar
  • 60.1k
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 ...
Ronnie Pavlov's user avatar
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 $...
Ronnie Pavlov's user avatar
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 ...
Vaughn Climenhaga's user avatar
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 ...
Ilkka Törmä's user avatar
6 votes
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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 = ...
Vaughn Climenhaga's user avatar
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 ...
Ville Salo's user avatar
  • 6,337
5 votes
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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 ...
Deinst's user avatar
  • 376
5 votes
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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 $(f^{-n}P)_{n\ge 0}$ ...
YCor's user avatar
  • 60.1k
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,\...
Anthony Quas's user avatar
  • 22.5k
4 votes

Convex combinations of Bernoulli Measures

Maybe it is good to note that a similar question: What is the weak-$^*$ closure of the set $$ D=\{\mu\in\mathcal{M}_\sigma: \mu\text{ is isomorphic to some }\nu\in C\}? $$ has a dramatically different ...
Dominik Kwietniak's user avatar
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 ...
Anthony Quas's user avatar
  • 22.5k
4 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 ...
Norbert Schuch's user avatar
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 ...
Benjamin Steinberg's user avatar
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'...
Anthony Quas's user avatar
  • 22.5k
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 ...
Ville Salo's user avatar
  • 6,337
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. ...
Surb's user avatar
  • 662
4 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 ...
Ville Salo's user avatar
  • 6,337
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 $x, x' \in X$...
Sophie M's user avatar
  • 675
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,...
Ilkka Törmä's user avatar

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