Dtjytyk

After Brexit, will the EU recognize British passports that are valid for more than ten years?

Did Amazon pay $0 in taxes last year?

Too soon for a plot twist?

I am the light that shines in the dark

Professor forcing me to attend a conference, I can't afford even with 50% funding

Limpar string com Regex

Why would /etc/passwd be used every time someone executes `ls -l` command?

What is the orbit and expected lifetime of Crew Dragon trunk?

Vector-transposing function

Is this Paypal Github SDK reference really a dangerous site?

Should I file my taxes? No income, unemployed, but paid 2k in student loan interest

How can I portion out frozen cookie dough?

The (Easy) Road to Code

Unfamiliar notation in Diabelli's "Duet in D" for piano

How to make sure I'm assertive enough in contact with subordinates?

Rationale to prefer local variables over instance variables?

Generating a list with duplicate entries

Why do we say 'Pairwise Disjoint', rather than 'Disjoint'?

Where is the License file location for Identity Server in Sitecore 9.1?

Can I challenge the interviewer to give me a proper technical feedback?

How to install "rounded" brake pads

Can I negotiate a patent idea for a raise, under French law?

How to educate team mate to take screenshots for bugs with out unwanted stuff

Why does a car's steering wheel get lighter with increasing speed

Behaviour of two distinct, arbitrary points of the Thue-Morse Subshift

Nice corollaries to Poincaré-Bendixson theoremBi-infinite sequences of $0$'s and $1$'sOrbit of shift operator in symbolic dynamicsShowing that a family of metrics induce all the same topology on special sequence spaceElementary properties of gradient systemsErgodic measure for the left shift supported on an orbitA group action on an infinite set with the cofinite topology$omega(x_0)=mathcal{O}(x_0)$ for periodic points $x_0$ergodic system has dense orbits a.e., continuity of TMinimal dynamical systems in $2^{mathbb N}$

3

$begingroup$

Let $x^+$ denote the infinite Thue-Morse word, $$x^+ = 0110100110010110ldots,$$ which is defined as being the only fixed point starting with $0$ of the morphism $S$ (defined over the words on the alphabet ${0,1}$) given by $S(0) = 01$ and $S(1) = 10$ - that is, $$x^+ = S^{omega}(0) = limlimits_{n geqslant 1} S^n(0);$$ this is only one among many different, equivalent definitions of such word (for instance, pick $A_0 = 0$ and let $A_{n+1} = A_n,^frownoverline{A_{n}}$ - where $^frown$ denotes concatenation and $overline{A_n}$ denotes the bitwise complement of $A_n$. Then $x^+ = limlimits_{n in mathbb{N}} A_n$). From now on, let $x^+ = (t_n)_{n in mathbb{N}}$.

It is known that the Thue-Morse word is uniformly recorrent (meaning, for every finite subword $w$ of $x^+$ there is some $n in mathbb{N}$ such that, for every $i in mathbb{N}$, the block $t_{i + 1}t_{i + 2}ldots t_{i + n}$ contains some ocurrence of $w$) but it is $mathbf{textrm{not}}$ ultimately periodic (meaning, it is not true that there are $p geqslant 1$ and $N geqslant 0$ such that $t_i = t_{i + p}$ for all $i geqslant N$).

The two-sided Thue Morse sequence $x = (x_i)_i in mathbb{Z}$ is defined as $x = x^{-},^frown x^+$, meaning that $x_n = t_n$ for all $n geqslant 0$ and $x_{-n} = t_{n - 1}$ for all $n geqslant 1$. So, $$x = ldots 100101100.1101001 ldots$$

Let $sigma$ denote (as usual) the left shift map on $^mathbb{Z}{0,1}$. The Thue-Morse subshift $X$ is the closure (in the Tychonoff topology) of the orbit ${sigma^n(x): n geqslant 0}$. Equivalently, $X$ is the family of all doubly infinite sequences $z$ such that every finite subword of $z$ is a subword of $x^+$. It is known that $X$ is a infinite, minimal subshift (that is, every point of $X$ has a dense orbit in $X$).

My question is:

Let $s = (s_i)_{i in mathbb{Z}},$ and $u = (u_i)_{i in mathbb{Z}},$ be two distinct points of the Thue-Morse subshift $X$ - that is, there is some integer $i in mathbb{Z}$ such that $s_i neq u_i$. Consider the set of integers given by $${i in mathbb{Z}: s_i neq u_i }.$$ Is it true that such set is, necessarily, an $mathbf{textrm{infinite}}$ set of integers ?

My conjecture is that the answer is "yes", but I wasn't able to produce a proof.

dynamical-systems

share|cite|improve this question

edited yesterday

Samuel G. Silva

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716

$endgroup$

add a comment | 

3

$begingroup$

Let $x^+$ denote the infinite Thue-Morse word, $$x^+ = 0110100110010110ldots,$$ which is defined as being the only fixed point starting with $0$ of the morphism $S$ (defined over the words on the alphabet ${0,1}$) given by $S(0) = 01$ and $S(1) = 10$ - that is, $$x^+ = S^{omega}(0) = limlimits_{n geqslant 1} S^n(0);$$ this is only one among many different, equivalent definitions of such word (for instance, pick $A_0 = 0$ and let $A_{n+1} = A_n,^frownoverline{A_{n}}$ - where $^frown$ denotes concatenation and $overline{A_n}$ denotes the bitwise complement of $A_n$. Then $x^+ = limlimits_{n in mathbb{N}} A_n$). From now on, let $x^+ = (t_n)_{n in mathbb{N}}$.

It is known that the Thue-Morse word is uniformly recorrent (meaning, for every finite subword $w$ of $x^+$ there is some $n in mathbb{N}$ such that, for every $i in mathbb{N}$, the block $t_{i + 1}t_{i + 2}ldots t_{i + n}$ contains some ocurrence of $w$) but it is $mathbf{textrm{not}}$ ultimately periodic (meaning, it is not true that there are $p geqslant 1$ and $N geqslant 0$ such that $t_i = t_{i + p}$ for all $i geqslant N$).

The two-sided Thue Morse sequence $x = (x_i)_i in mathbb{Z}$ is defined as $x = x^{-},^frown x^+$, meaning that $x_n = t_n$ for all $n geqslant 0$ and $x_{-n} = t_{n - 1}$ for all $n geqslant 1$. So, $$x = ldots 100101100.1101001 ldots$$

Let $sigma$ denote (as usual) the left shift map on $^mathbb{Z}{0,1}$. The Thue-Morse subshift $X$ is the closure (in the Tychonoff topology) of the orbit ${sigma^n(x): n geqslant 0}$. Equivalently, $X$ is the family of all doubly infinite sequences $z$ such that every finite subword of $z$ is a subword of $x^+$. It is known that $X$ is a infinite, minimal subshift (that is, every point of $X$ has a dense orbit in $X$).

My question is:

Let $s = (s_i)_{i in mathbb{Z}},$ and $u = (u_i)_{i in mathbb{Z}},$ be two distinct points of the Thue-Morse subshift $X$ - that is, there is some integer $i in mathbb{Z}$ such that $s_i neq u_i$. Consider the set of integers given by $${i in mathbb{Z}: s_i neq u_i }.$$ Is it true that such set is, necessarily, an $mathbf{textrm{infinite}}$ set of integers ?

My conjecture is that the answer is "yes", but I wasn't able to produce a proof.

dynamical-systems

share|cite|improve this question

edited yesterday

Samuel G. Silva

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716

$endgroup$

add a comment | 

$begingroup$

Let $x^+$ denote the infinite Thue-Morse word, $$x^+ = 0110100110010110ldots,$$ which is defined as being the only fixed point starting with $0$ of the morphism $S$ (defined over the words on the alphabet ${0,1}$) given by $S(0) = 01$ and $S(1) = 10$ - that is, $$x^+ = S^{omega}(0) = limlimits_{n geqslant 1} S^n(0);$$ this is only one among many different, equivalent definitions of such word (for instance, pick $A_0 = 0$ and let $A_{n+1} = A_n,^frownoverline{A_{n}}$ - where $^frown$ denotes concatenation and $overline{A_n}$ denotes the bitwise complement of $A_n$. Then $x^+ = limlimits_{n in mathbb{N}} A_n$). From now on, let $x^+ = (t_n)_{n in mathbb{N}}$.

It is known that the Thue-Morse word is uniformly recorrent (meaning, for every finite subword $w$ of $x^+$ there is some $n in mathbb{N}$ such that, for every $i in mathbb{N}$, the block $t_{i + 1}t_{i + 2}ldots t_{i + n}$ contains some ocurrence of $w$) but it is $mathbf{textrm{not}}$ ultimately periodic (meaning, it is not true that there are $p geqslant 1$ and $N geqslant 0$ such that $t_i = t_{i + p}$ for all $i geqslant N$).

The two-sided Thue Morse sequence $x = (x_i)_i in mathbb{Z}$ is defined as $x = x^{-},^frown x^+$, meaning that $x_n = t_n$ for all $n geqslant 0$ and $x_{-n} = t_{n - 1}$ for all $n geqslant 1$. So, $$x = ldots 100101100.1101001 ldots$$

Let $sigma$ denote (as usual) the left shift map on $^mathbb{Z}{0,1}$. The Thue-Morse subshift $X$ is the closure (in the Tychonoff topology) of the orbit ${sigma^n(x): n geqslant 0}$. Equivalently, $X$ is the family of all doubly infinite sequences $z$ such that every finite subword of $z$ is a subword of $x^+$. It is known that $X$ is a infinite, minimal subshift (that is, every point of $X$ has a dense orbit in $X$).

My question is:

Let $s = (s_i)_{i in mathbb{Z}},$ and $u = (u_i)_{i in mathbb{Z}},$ be two distinct points of the Thue-Morse subshift $X$ - that is, there is some integer $i in mathbb{Z}$ such that $s_i neq u_i$. Consider the set of integers given by $${i in mathbb{Z}: s_i neq u_i }.$$ Is it true that such set is, necessarily, an $mathbf{textrm{infinite}}$ set of integers ?

My conjecture is that the answer is "yes", but I wasn't able to produce a proof.

dynamical-systems

share|cite|improve this question

edited yesterday

Samuel G. Silva

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716

$endgroup$

share|cite|improve this question

edited yesterday

Samuel G. Silva

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716

share|cite|improve this question

edited yesterday

Samuel G. Silva

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716

asked Sep 19 '17 at 17:54

Samuel G. SilvaSamuel G. Silva

716