Quasiconformal contact foliations

We show that every quasiconformal contact foliation supports an invariant metric and characterise such foliations by the dynamical property of $C^1$-equicontinuity. We prove that a generalisation of the Weinstein conjecture holds for quasiconformal contact foliations, and provide a lower bound to the number of closed leaves. In particular, we show that the Weinstein conjecture holds for quasiconformal Reeb fields.