  We exhibit a continuously varying family $F_\lambda$ of homeomorphisms of the
sphere $S^2$, for which each $F_\lambda$ is a measurable pseudo-Anosov map.
Measurable pseudo-Anosov maps are generalizations of Thurston's pseudo-Anosov
maps, and also of generalized pseudo-Anosov maps. They have a transverse pair
of invariant full measure turbulations, consisting of streamlines which are
dense injectively 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.
  Each map $F_\lambda$ is semi-conjugate to the inverse limit of the core tent
map with slope $\lambda$: it is topologically transitive, ergodic with respect
to a background Oxtoby-Ulam measure, has dense periodic points, and has
topological entropy $h(F_\lambda) = \log \lambda$ (so that no two $F_\lambda$
are topologically conjugate). For a full measure, dense $G_\delta$ set of
parameters, $F_\lambda$ is a measurable pseudo-Anosov map but not a generalized
pseudo-Anosov map, and its turbulations are nowhere locally regular.
