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I am wondering if there is an example $(f,X)$ and $n\ge2$ with $\Omega(f)\neq\Omega(f^n)$. It is interesting to know such examples.

There is an observation that for a homeo $f:X\to X$, if $x\in\omega(x,f)$, then $x\in\omega(x,f^n)$ for each $n\ge1$. The proof is:

Let $n\ge2$ be given. Note that $\omega(x,f)=\bigcup_{0\le k< n}\omega(f^kx,f^n)$. So there exists $k$ with $x\in\omega(f^kx,f^n)$.

If $k=0$ we are done.

Otherwise let $l=n-k\in[1,n-1]$. Then $f^lx\in\omega(x,f^n)$. We show inductively $f^{jl}x\in\omega(x,f^n)$ for each $j\ge1$. Since $\omega(x,f^n)$ is strictly $f^n$-invariant and $f^{nl}x\in\omega(x,f^n)$, we get $x\in\omega(x,f^n)$, too.

$f^{(j+1)l}x=f^l(f^{jl}x)\in f^l\omega(x,f^n)=\omega(f^lx,f^n)\subset\omega(x,f^n)$, where $\in$ is from induction hypothesis and $\subset$ is from the forward invariance of $\omega$-sets.

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I am wondering if there is an example $(f,X)$ and $n\ge2$ with $\Omega(f)\neq\Omega(f^n)$. It is interesting to know such examples.

There is an observation that for a homeo $f:X\to X$, if $x\in\omega(x,f)$, then $x\in\omega(x,f^n)$ for each $n\ge1$. The proof is:

Let $n\ge2$ be given. Note that $\omega(x,f)=\bigcup_{0\le k< n}\omega(f^kx,f^n)$. So there exists $k$ with $x\in\omega(f^kx,f^n)$.

If $k=0$ we are done.

Otherwise let $l=n-k\in[1,n-1]$. Then $f^lx\in\omega(x,f^n)$. We show inductively $f^{jl}x\in\omega(x,f^n)$ for each $j\ge1$. Since $\omega(x,f^n)$ is strictly $f^n$-invariant and $f^{nl}x\in\omega(x,f^n)$, we get $x\in\omega(x,f^n)$, too.

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I am wondering if there is an example $(f,X)$ and $n\ge2$ with $\Omega(f)\neq\Omega(f^n)$. It is interesting to know.