Entropy and dynamics of $C^1$-foliations
by Steven Hurder

HISTORY:

Shortly after the geometric entropy h_g(F) of a $C^1$-foliation F was introduced by Ghys-Langevin-Walczak, Hurder wrote the paper Ergodic theory of foliations and a theorem of Sacksteder which laid out a program for the study of the smooth dynamics of foliations, using the technique of studying the dynamics of the foliation geodesic flow relative to the leaves of the foliation. Parts of this program have been published in the intervening period by the author and others, though many of the claims regarding the relation between foliation geometric entropy and smooth dynamics have not been written up.

The current paper resulted when Larry Conlon raised this issue again, in response to a talk on the recent work of Hurder & Langevin: is it possible to give a proof that positive entropy implied the existence of resilient leaves for codimension one foliations using ergodic techniques. A proof of this was worked out in December 1999, and this along with a previous result (Theorem 2 of the abstract) is the heart of this paper which was written up in August 2000.