Unimodal Measurable Pseudo-Anosov Maps

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 the generalized pseudo-Anosov maps of [19]. 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.