Abstract

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 (Geom. Topol. 8 (2004), 1127–1188). 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 semiconjugate 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 $h(F_{\lambda })=\log <!--FUNCTION\; APPLICATION-->\lambda$ (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.

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