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.


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


[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.)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.