Branched surfaces and Markov minimal sets in foliations of 3-manifolds

Abstract

We find necessary and sufficient conditions for the existence of a Markov minimal set in a transversely orientable codimension one foliation of a closed Riemannian 3-manifold M. Specifically, we show that if a foliation F has a Markov minimal set, then any branched surface carrying F can be modified to contain a 1-dimensional graph that captures the holonomy of F and has a specific structure. Conversely, when a branched surface W contains a graph with these properties, the holonomy group for any foliation carried by W contains a Markov sub-pseudgroup; this together with an added criterion are sufficient for the existence of a Markov minimal set in any foliation F carried by W. When the sufficiency criteria are satisfied, the Markov minimal set is stable under C1-perturbations of F, provided that the complement of W in M is simply connected.