We prove that a real analytic pseudo-rotation f of the disc or the sphere is never topologically mixing. When the rotation number of f is of Brjuno type, the latter follows from a KAM theorem of Rüssmann on the stability of real analytic elliptic fixed points. In the non-Brjuno case, we prove that a pseudo-rotation of class C k , k ≥ 2, is C k−1 -rigid using the simple observation, derived from Franks' Lemma on free discs, that a pseudo-rotation with small rotation number compared to its C 1 (or Hölder) norm must be close to Identity.From our result and a structure theorem by Franks and Handel (on zero entropy surface diffeomorphisms) it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.In our proof we need an a priori limit on the growth of the derivatives of the iterates of a pseudo-rotation that we obtain via an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f , that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential.

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The convex reflective diffraction grating is an essential optical component that lends itself to various applications. In this work, we first outline the design principles of convex diffraction gratings from wavefront quality and efficiency perspectives. We then describe a unique fabrication method that allows for the machining of convex diffraction gratings with variable groove structure, which is extendable to rotationally non-symmetric convex diffraction grating substrates. Finally, we demonstrate two quantitative wavefront measurement methods and respective experimental validation.

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle, the smooth centralizer is either virtually Z or contains a smooth flow.At the heart of this work are two very different rigidity phenomena. The first was discovered in [2, 3]: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The second phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25].We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer.

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