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!

To rpotrie: I am looking for sufficient conditions on the spaces (say, manifolds) and the maps (say, the regularity and domination or partial hyperbolicity) 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.

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!

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.

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!

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!

To rpotrie: I am looking for sufficient conditions on the spaces (say, manifolds) and the maps (say, the regularity and domination or partial hyperbolicity) 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.