Generating positive geometric entropy from recurrent leaves

TL;DR

This paper introduces a new perturbation method for $C^r$-foliations by surfaces, enabling the creation of resilient leaves from recurrent leaves, thus bridging complex and trivial dynamics in nearby foliations.

Contribution

The paper develops a $C^r$-perturbation technique in the Epstein topology to generate resilient leaves from recurrent leaves in surface foliations.

Findings

Established a perturbation method to produce resilient leaves

Demonstrated how recurrence can lead to complex dynamics

Constructed examples of close foliations with differing dynamics

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 $\mathcal F$ to obtain a foliation $\mathcal G$, $C^r-$close to $\mathcal F$ 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).