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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
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[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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