ds.dynamical systems - Is the composition of non-wandering maps still non-wandering?

I am wondering if there is an example $(f,X)$ and $nge2$ with $Omega(f)neqOmega(f^n)$.
It is interesting to know such examples.

There is an observation that for a homeo $f:Xto X$, if $xinomega(x,f)$, then $xinomega(x,f^n)$ for each $nge1$. The proof is:

Let $nge2$ be given. Note that $omega(x,f)=bigcup_{0le k< n}omega(f^kx,f^n)$. So there exists $k$ with $xinomega(f^kx,f^n)$.

If $k=0$ we are done.

Otherwise let $l=n-kin[1,n-1]$. Then $f^lxinomega(x,f^n)$. We show inductively $f^{jl}xinomega(x,f^n)$ for each $jge1$. Since $omega(x,f^n)$ is strictly $f^n$-invariant and $f^{nl}xinomega(x,f^n)$, we get $xinomega(x,f^n)$, too.

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