Absolutely Singular Dynamical Foliations

Let A ³ be the product of the automorphism of T ² and of the identity on T ¹. A small perturbation g of A ³ among volume preserving diffeomorphisms will have an invariant family of smooth circles Γ forming a continuous foliation of T ³. Corresponding to the vector bundle tangent to the circles Γ there is a “central” Lyapunov exponent of (g, volume), which is nonzero for an open set of ergodic g's. This surprising result of Shub and Wilkinson is complemented here by showing that the volume on T ³ has atomic conditional measures on the Γ's: there is a finite k such that almost every Γ carries $k$ atoms of mass 1/k. Received: 26 September 2000 / Accepted: 8 December 2000