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Let $f:M \rightarrow M$ be a partially hyperbolic diffeomorphism of $M$ with the usual definition that at each $p$ tangent space splits to $Df$ invariant subspaces: $T_pM = E^s_p \oplus E^c_p \oplus E^u_p$ with the relevant constants on the norm of $Df$ restricted to these spaces. I just need one which is $\mu^n < |Df^n_{E^u}|$.

I have some confusions about how people define the center-stable manifolds etc. In the case of just partially hyperbolic fixed point $p$ in $R^n$, Robinson in his book defines the center stable manifold of the fixed point as $$ W^{cs}(p) = \{x | \lim_{n\to \infty}\frac{1}{\mu^n}d(f^n(x),p)=0\} $$ (after extending the map to whole of $\mathbb{R}^n$, if its not already defined everywhere) and says that although if not unique, such center manifolds exists and are tangent to $E^c$. In the case of partially hyperbolic diffeomorphism of manifolds, I have never seen this characterization. They basically define $W^{cs}(p)$ as a submanifold tangent to $E^{cs}$.

My question is for a partially hyperbolic set. Is it not necessary that a submanifold $N(p)$ passing through $p$ which satisfies $$N(p) = \{x | \lim_{n\rightarrow \infty}\frac{1}{\mu^n}d(f^n(x),p)=0\} $$ will be tangent to $E^c$? I was able to show that the unstable manifold has to be transversal to this manifold but transversality is not enough to show that tangent space of this manifold has to be $E^c \oplus E^s$

Why is it not preferred to define central stable behaviour in this manner for this setting? Is there also a center stable manifold theorem for this setting in the manner which says, for all $p$ there is always a submanifold passing through $p$, which satisfies the convergence criteria I wrote above and is tangent to $E^s \oplus E^c$?

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The reason is simply that the "right" rates for a partially hyperbolic set may depend on $p$. Of course, since you tell us in your second paragraph that you only rates of a certain type, then it may be fine to use your second "definition".

A second aspect, more important, but also more technical, concerns the regularity of the manifolds and that depends on somewhat delicate relations between your rate $\mu$ and other measuring the top and bottom rates on each fiber.

Up to these points, the answer to your last question is yes.

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