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Let be $\Lambda$ a hosrseshoe for a $C^r$ ($r\geq2$) dipheomorphism $f:M\to M$ where $M$ is a two dimensional manifold. A well-known result is the following:

for each $x\in \Lambda$ let be $W^u(x)$ and $W^s(x)$ the unstable and stable manifolds respectively of $f$ in $x$. Then exists neighborhoods $\mathcal{V}_1$ and $\cal{V}_2$ of $\Lambda$ where are defined $C^1$ foliations $\cal{F}_1$ and $\cal{F}_2$ such that $T_p\mathcal{F}_1=E^u_p$ and $T_p\mathcal{F}_2=E^s_p$ where $T_p\mathcal{F}_i$ denotes the tangent space on the leave $\cal{F}_i$.

My question is: how construct this foliations ?

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See this question:… – Vaughn Climenhaga Dec 19 '10 at 1:53
To summarise the answer from that question... the argument is on p. 166 of Palis and Takens' book:… – Vaughn Climenhaga Dec 19 '10 at 1:54
Hi Vaughn! I knew this reference, but it does not say much for me beyond the result itself. I'm looking for something more concrete. I want to believe that this result is not just part of folklore. Sorry if I do not understand out of ignorance. – user11178 Dec 19 '10 at 12:46
There are a few typos in your title and question body. – Mariano Suárez-Alvarez Jan 3 '11 at 19:27
Yeah, there seem to be typos; its diffeomorphism, right? – drbobmeister Jan 11 '11 at 2:56
up vote 0 down vote accepted

You can find an alternative proof (to the one of Palis Takens) in section 6.4 of this survey.

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