## On a regularity problem occurring in connection with Anosov diffeomorphisms.(English)Zbl 0603.58019

Let $${\mathcal M}$$ be a $$C^{\infty}$$-manifold and $${\mathcal F}_ s$$ and $${\mathcal F}_ u$$ be two Hölder foliations, transverse, and with uniformly $$C^{\infty}$$ leaves. If a function f is uniformly $$C^{\infty}$$ along the leaves of the two foliations, then it is $$C^{\infty}$$ on $${\mathcal M}$$. The proof is elementary.

### MSC:

 37D99 Dynamical systems with hyperbolic behavior 57R30 Foliations in differential topology; geometric theory

### Keywords:

Hölder foliations; $$C^{\infty }$$ leaves
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### References:

 [1] Campanato, S.: Proprietá dí una famiglia dí spazi funzionali. Ann. Sc. Norm. Super. Pisa18, 137-160 (1964) · Zbl 0133.06801 [2] de la Llave, R.: Invariants for smooth conjugacy of hyperbolic systems. II (submitted to Commun. Math. Phys.) · Zbl 0673.58036 [3] de la Llave, R., Marco, J.M., Moriyon, R.: Canonical perturbation theories for Anosov’s systems and regularity results for the Livsic’s Cohomology Equation. Ann. Math. to appear. See also Bull. Am. Math. Soc.12, 1, 91-94 (1985) · Zbl 0576.58011 [4] Marco, J.M., Moriyon, R.: Invariants for smooth conjugacy of hyperbolic systems. I (submitted to Commun. Math. Phys.)
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