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‎let ‎‎$‎f:M‎\rightarrow ‎M‎$ ‎be a‎ ‎‎‎‎diffeomrphism and‎‎‎ ‎$ ‎\Lambda‎ $ a‎‎ ‎hyperbolic ‎set. ‎We ‎can ‎give a‎ ‎characterization ‎of ‎(local) ‎stable ‎and ‎unstable ‎manifolds ‎by‎: ‎‎for ‎‎‎‎$‎\varepsilon‎>0‎‎‎$ ‎small ‎enough‎ ‎$‎W‎^{s}‎_{‎\varepsilon‎}(x)=‎\lbrace ‎y‎\vert ‎d(‎f‎^{n}(x), ‎f‎^{n}(y))‎\leq‎‎‎ ‎\varepsilon ‎for ‎n‎\geq0‎‎ ‎\rbrace‎‎$‎‎

‎$W‎^{s}(x)=‎\lbrace y‎\vert d(f‎^{n}(x), f‎^{n}(y))‎\rightarrow 0‎\rbrace‎‎‎‎‎‎‎‎‎$‎‎‏‎

and same for unstable manifold.‎ My question is if ‎$ ‎\Lambda‎ $ ‎is ‎partially ‎hyperbolic ‎set ‎for f‎, ‎then ‎in this case what is the definition of stable and unstable manifolds and what is there characterizations?

‎ you ‎can ‎find ‎the ‎definitions ‎of ‎hyperbolic ‎and ‎partial ‎hyperbolic ‎sets ‎and ‎more ‎here‎‎

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Welcome to Math.SE! –  user93957 Nov 29 '13 at 18:32

1 Answer 1

The notions you probably want are those of strongly stable and central stable manifolds.

Consider a discrete dynamical system $f : \mathbb{R}^n \to \mathbb{R}^n$ with a fixed point at the origin (it is not hard to generalize this to the partially hyperbolic case, or even to the nonuniformly partially hyperbolic case, if you like). Let $T = df_0$, and that $T$ has eigenvalues both inside and outside the unit circle (and maybe some on the unit circle, too!).

Then, I can consider a set of the form $$ W^{ss}_{\alpha, \epsilon} = \{x \in \mathbb{R}^n \mid |f^n x| \leq \epsilon \alpha^n, n \geq 0\} $$ where we assume that some of the eigenvalues of $T$ have modulus smaller than $\alpha > 0$, and $\epsilon > 0$ is considered very small. Then, the set $W^{ss}_{\alpha, \epsilon}$ actually has the same kind of structure as did the stable manifold in the case when $T$ was hyperbolic; the only difference here is that it consists of those points which approach $0$ at a certain minimum exponential rate $\alpha$.

The stable manifold is tangent to the stable subspace of the linearization at a point; what would the analogous statement for strongly stable manifolds be?

Central stable manifolds are invariant manifolds which are tangent to stable directions as well as some central (and even expanding) directions, but these are harder to deal with, usually nonunique, and less regular than their strongly stable counterparts.

For more information, you can check out the classic Hirsch, Pugh, & Schub, or the classnotes of O. Lanford at this site.

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