  We study the dynamics of measurable pseudo-Anosov homeomorphisms of surfaces,
a generalization of Thurston's pseudo-Anosov homeomorphisms. A measurable
pseudo-Anosov map has a transverse pair of full measure turbulations consisting
of streamlines which are dense immersed lines: these turbulations are equipped
with measures which are expanded and contracted uniformly by the homeomorphism.
The turbulations need not have a good product structure anywhere, but have some
local structure imposed by the existence of tartans: bundles of unstable and
stable streamline segments which intersect regularly, and on whose
intersections the product of the measures on the turbulations agrees with the
ambient measure. We prove that measurable pseudo-Anosov maps are transitive,
have dense periodic points, sensitive dependence on initial conditions, and are
ergodic with respect to the ambient measure. Measurable pseudo-Anosovs maps
were introduced in [11], where we constructed a parameterized family of
non-conjugate examples on the sphere.
