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

ABSTRACT:

The geometric entropy h_g(F) of a $C^1$-foliation F introduced by Ghys-Langevin-Walczak is a measure of the complexity of the leaf dynamics. There are three main results to date relating the geometric entropy to the dynamical complexity of a foliation:

1. For F a transversally $C^2$-foliation of codimension-1 with h_g(F) > 0, F has a resilient leaf (Ghys-Langevin-Walczak)
2. For F a $C^1$-foliation of arbitrary codimension with h_g(F) = 0 F has a non-trivial holonomy invariant transverse measure (Ghys-Langevin-Walczak).
3. For F a $C^1$-foliation of arbitrary codimension with h_g(F) = 0, every leaf of F must have zero entropy as complete metric space (Attie-Hurder).

The purpose of this paper is to establish two new results relating the geometric entropy of a foliation with its dynamics:

THEOREM 1: For F a transversally $C^1$-foliation of codimension one with h_g(F) > 0, F has a resilient leaf.

The proof of Theorem 1 uses methods similar to techniques of classical ergodic theory, invoking counting arguments and properties of the foliation geodesic flow. The proof is thus fundamentally different than that of Ghys-Langevin-Walczak, which used delicate properties of the structure theory of $C^2$-foliations of codimension one.

THEOREM 2: For F a $C^{1+a}$-foliation of arbitrary codimension with h_g(F) > 0, then there exists a leafwise geodesic path so that the transerse holonomy along the path admits a stable transverse manifold which is (transversally) attracted to the leaf at an exponential rate.

There is a natural extension of the notion of a distal group action to foliations, and Theorem 2 implies:

COROLLARY: If F is a distal $C^{1+a}$-foliation, then h_g(F) = 0.