We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala’s conjecture that the Hausdorff dimension of a$k$-quasicircle is at most 1+$k$^{2}.

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.

We will prove Sarnak’s conjecture on Möbius disjointness for continuous interval maps of zero entropy and also for orientation-preserving circle homeomorphisms by reducing these result to a well-known theorem of Davenport from 1937.