American Institute of Mathematical Sciences

Let \begin{document}$\mathscr{H}$\end{document} denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on \begin{document}$\mathbb{R}^2$\end{document} is in \begin{document}$L^2(\mathscr{H}, \mu)$\end{document}, where \begin{document}$\mu$\end{document} is the Lebesgue measure on \begin{document}$\mathscr{H}$\end{document}, and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to \begin{document}$SL(2,\mathbb{R})$\end{document}-invariant measures on strata satisfying certain integrability conditions.

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 \begin{document}$ SU(p,q) $\end{document} type are realized by appropriate covering constructions.

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 \begin{document}$ X $\end{document}, and consider the measures on \begin{document}$ X $\end{document} coming from averaging the uniform measures on all the saddle connections of length at most \begin{document}$ R $\end{document}. Then, as \begin{document}$ R\to\infty $\end{document}, the weak limit of these measures exists and is equal to the area measure on \begin{document}$ X $\end{document} coming from the flat metric. This implies that, on a rational-angled billiard table, the billiard trajectories that start and end at a corner of the table are equidistributed on the table. We also show that any weak limit of a subsequence of the counting measures on \begin{document}$ S^1 $\end{document} given by the angles of all saddle connections of length at most \begin{document}$ R_n $\end{document}, as \begin{document}$ R_n\to\infty $\end{document}, is in the Lebesgue measure class. The proof of the equidistribution result uses the angle result, together with the theorem of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.

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 \begin{document}$ \theta $\end{document} on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if \begin{document}$ \tan\theta $\end{document} has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and \begin{document}$ \tan\theta $\end{document} has bounded partial quotients, the square-tiled interval exchange transformation \begin{document}$ T $\end{document} is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.

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.

For all \begin{document}$ d $\end{document} belonging to a density-\begin{document}$ 1/8 $\end{document} subset of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group \begin{document}$ \mathrm{SO}^*(2d) $\end{document} in its standard representation as the Zariski-closure of a factor of its monodromy. We prove that this conjecture holds for the first elements of this subset, showing that the group \begin{document}$ \mathrm{SO}^*(2d) $\end{document} is realizable for every \begin{document}$ 11 \leq d \leq 299 $\end{document} such that \begin{document}$ d = 3 \bmod 8 $\end{document}, except possibly for \begin{document}$ d = 35 $\end{document} and \begin{document}$ d = 203 $\end{document}.

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 \begin{document}$ \theta $\end{document} on a translation surface and we study the asymptotic behavior for \begin{document}$ r\to 0 $\end{document} of the time needed by orbits to hit the \begin{document}$ r $\end{document}-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction \begin{document}$ \theta $\end{document}. In higher genus, we consider a generalized geometric notion of Diophantine type of a direction \begin{document}$ \theta $\end{document} and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.