Journal of Modern Dynamics
2019 , Volume 14
William A. Veech Memorial Volume edited by Jon Chaika, Giovanni Forni, Howard Masur, and John Smillie
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We describe the Kontsevich–Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich–Zorich monodromies of
We show that Sarnak's conjecture on Möbius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.
Fix a translation surface
We introduce a class of objects which we call 'dilation surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalize the notion of Veech group to dilation surfaces, and we prove a structure result about these Veech groups.
We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction
We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.
We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. These objects are exponentially attracted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is attuned to the geometry of the surface and its deformations.
We provide a criterion for a point satisfying the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with discrete spectrum.
We consider the flow in direction
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