Generating positive geometric entropy from recurrent leaves

Abstract

In this paper we introduce a new C r −perturbation procedure, with respect to the C r -Epstein topology, for C r -foliations by surfaces. Using this perturbation procedure we show how one can use the existence of recurrent leaves of certain C r −foliation F to obtain a foliation G, C r −close to F in the C r −Epstein topology, which has a resilient leaf. In particular, one can take advantage of recurrence property to construct examples of C r −foliations by surfaces, C r −close to each other and such that one of them has a resilient leaf while the other is Riemannian (therefore has trivial dynamics).