nonwandering set and Birkhoff center

Question

I am wondering if there are some results about the depth of a diffeomorphism on a manifold.

More precisely, $(M,f)$ be a diffeomorphism. For each compact invariant subset $E$, let $\Omega(f, E)$ be the nonwandering subset of $f$ relative to $E$. Let $\Omega_1=\Omega(f,M)$, $\Omega_{n+1}=\Omega(f,\Omega_n)$, and $\Omega_a=\cap\Omega_b$ over $b < a$ for a limit cardinal $b$... etc

So my question is; are there some conditions under which the diffeo has a finite depth, that is, $\Omega_{n+1}=\Omega_n$ for some $n$?

There are examples of topological systems with countable depths. I do not know what can happen in the smooth category. I googled and found that the depths of circle maps or interval maps are less than 2.

Thanks!

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To rpotrie: I am looking for sufficient conditions on the spaces (say, manifolds) and the maps (say, the regularity) such that $f$ has finite center depth. As rpotrie mentioned, Axiom A maps (hence all Anosov) always have center depth 1, the maps with $\Omega(f)$ hyperbolic have center depth less than 2.

For example partially hyperbolicity may not be a good candidate since the direct product $f\oplus g:M\times N\to M\times N$ has transfinite center depth if one of $f$ or $g$ has.

Answer 1

There are some results which guaranty that the depth is $1$, but I don't know if they are quite what you are looking for.

First, in the case of Homeomorphisms and $C^1$-diffeomorphisms of a Manifold, the famous Closing Lemma implies that generic homeomorphisms (resp. generic $C^1$-diffeomorphisms) verify that the set of periodic points is dense in the non-wandering set, and thus, the depth is one.

Also, when the map has some hyperbolic conditions, one knows that in surfaces the nonwandering set must be the closure of hyperbolic points. This is a Theorem by Newhouse and Palis which says that: If $f$ is a surface diffeomorphism and the nonwandering set is hyperbolic, then, it is the closure of its periodic points (and thus, the depth is one).

In higher dimensions, there are examples by Dankner of diffeomorphisms whose nonwandering set is hyperbolic but the periodic points are not dense (in this hyperbolic case, this implies that the depth is $>1$). In this examples, the depth is exacly two, since for a hyperbolic set $\Lambda$, the periodic points are dense in $\Omega(f|_\Lambda)$.

Other of this results, the only knowledge I have is that examples where the depth is bigger than one are extremely patological (although I don't believe it is even known for $C^r$ topology with $r\geq 2$ if having depth one is a generic property, but maybe it can be proved that the depth is finite generically).