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can someone help me with this problem, I don't know how to start or what to do...

Let $Σ_2$ be the space of sequences of 0's & 1's and let $σ:Σ_2 \to Σ_2$ be the shift map given by: $$σ((t_1,t_2,t_3,...))=(t_2,t_3,t_4,...).$$ Let the point $t$ in $\Sigma_2$ be: $$t=(1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...)$$

Describe the set $ω(t,σ)$. In particular, show that this set is infinite numerable.

Describe the dynamics of $σ$ restricted to $ω(t,σ)$. In particular, show that $σ$ restricted to $ω(t,σ)$ doesn't contain dense orbits in $ω(t,σ)$.

Thanks so much for your help!

Martin Sleziak
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user392559
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1 Answers1

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The hard part of this is figuring out what $\omega(t,\sigma)$ looks like. I’ll do that and leave the rest to you for now.

Let $x\in\Sigma_2$ be any sequence with at least two ones that is not in the orbit of $t$. Then there are $k,\ell\in\Bbb Z^+$ such that $k<\ell$, $x_k=x_\ell=1$, $x_n=0$ for $k<n<\ell$, and either

  • $x_{2\ell-k+1}\ne 1$, or
  • $x_n=1$ for some $n$ such that $\ell<n<2\ell-k+1$.

In other words, $x$ has a substring

$$1,\underbrace{0,0,\ldots,0,0}_{\ell-k-1\text{ zeroes}},1$$

that is not immediately followed by

$$\underbrace{0,0,\ldots,0,0}_{\ell-k\text{ zeroes}},1\;,$$

as it would be in any member of the orbit of $t$. Then

$$\{y\in\Sigma_2:y_n=x_n\text{ for }k\le n\le2\ell-k+1\}$$

is an open nbhd of $x$ disjoint from the orbit of $t$. On the other hand, it’s straightforward to verify that every $x\in\Sigma_2$ with at most one $1$ is in $\omega(t,\sigma)$, so

$$\omega(t,\sigma)=\left\{\sigma^n(t):n\ge 0\right\}\cup\{z\}\cup\{e^{(n)}:n\in\Bbb Z^+\}\;,$$

where $z$ is the zero sequence, and $e^{(n)}=\langle\delta_{nk}:k\in\Bbb Z^+\rangle$ is the sequence whose $n$-th term is $1$, all other terms being $0$.

Brian M. Scott
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