I am reading the paper: The Hausdorff dimension of horseshoes of Diffeomorphism of surfaces, Bulletin Brazilian Mathematical Society, Ricardo Mañe,(1990). Roughly speaking the author state that, the unstable and stable foliation of horseshoe can be extended to a $C^1$ foliation on a neighborhood of the horseshoe. I can not locate a reference for this statement and I was not able to prove it. More precisely:

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$. Suppose that $\Lambda\subset M$ is a horseshoe for $f$. Why the stable and unstable foliations defined by $W^u(x)$ and $W^s(x)$, when $x$ varies $\Lambda$, extend to $C^1$ foliation of a neighborhood of $\Lambda$ ?

Could you help me with the argument or a reference for this statement ?


In page 166 of Palis-Takens book that is also stated. In the discusion before, the outline is given (see Appendix 1).


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