American Institute of Mathematical Sciences

We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain [3].

Using classical results of Rogers [12, Theorem 1] bounding the L²-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantitative Oppenheim theorem of Eskin-Margulis-Mozes [6] for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.

A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh extended Ratner's theorem to horocycle flows on hyperbolic surfaces of infinite area but finite genus. In this paper, we present the first joining classification result of a horocycle flow on a hyperbolic surface of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface.

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices \begin{document}$ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$\end{document}, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum \begin{document}$\mathscr{H}(1,1)$\end{document} of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold \begin{document}$M$\end{document}, pinched between two circle bundles whose ratio of radii is less than \begin{document}$\sqrt{2}$\end{document} carries either one short simple periodic orbit or carries at least cuplength \begin{document}$(M)+1$\end{document} simple periodic Reeb orbits.

Let \begin{document}$\varphi_0$\end{document} be a smooth area-preserving diffeomorphism of a compact surface \begin{document}$M$\end{document} and let \begin{document}$Λ_0$\end{document} be a horseshoe of \begin{document}$\varphi_0$\end{document} with Hausdorff dimension strictly smaller than one. Given a smooth function \begin{document}$f:M\to \mathbb{R}$\end{document} and a small smooth area-preserving perturtabion \begin{document}$\varphi$\end{document} of \begin{document}$\varphi_0$\end{document}, let \begin{document}$L_{\varphi, f}$\end{document}, resp. \begin{document}$M_{\varphi, f}$\end{document} be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of \begin{document}$f$\end{document} along the \begin{document}$\varphi$\end{document}-orbits of points in the horseshoe \begin{document}$Λ$\end{document} obtained by hyperbolic continuation of \begin{document}$Λ_0$\end{document}.

We show that, for generic choices of \begin{document}$\varphi$\end{document} and \begin{document}$f$\end{document}, the Hausdorff dimension of the sets \begin{document}$L_{\varphi, f}\cap (-∞, t)$\end{document} vary continuously with \begin{document}$t∈\mathbb{R}$\end{document} and, moreover, \begin{document}$M_{\varphi, f}\cap (-∞, t)$\end{document} has the same Hausdorff dimension as \begin{document}$L_{\varphi, f}\cap (-∞, t)$\end{document} for all \begin{document}$t∈\mathbb{R}$\end{document}.

Let \begin{document} $0<\lambda<1$ \end{document}. We consider the one-parameter family of circle \begin{document} $\lambda$ \end{document}-affine contractions \begin{document} $f_\delta:x \in [0,1) \mapsto \lambda x + \delta \; {\rm mod}\,1 $ \end{document}, where \begin{document} $0 \le \delta <1$ \end{document}. Let \begin{document} $\rho$ \end{document} be the rotation number of the map \begin{document} $f_\delta$ \end{document}. We will give some numerical relations between the values of \begin{document} $\lambda,\delta$ \end{document} and \begin{document} $\rho$ \end{document}, essentially using Hecke-Mahler series and a tree structure. When both parameters \begin{document} $\lambda$ \end{document} and \begin{document} $\delta$ \end{document} are algebraic numbers, we show that \begin{document} $\rho$ \end{document} is a rational number. Moreover, in the case \begin{document} $\lambda$ \end{document} and \begin{document} $\delta$ \end{document} are rational, we give an explicit upper bound for the height of \begin{document} $\rho$ \end{document} under some assumptions on \begin{document} $\lambda$ \end{document}.

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.

We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to \begin{document}$GL(2,\mathbb{R})$\end{document}-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle.

We construct an example of a Teichmüller geodesic ray whose limit set in the Thurston boundary of Teichmüller space is a \begin{document}$ d$\end{document}-dimensional simplex.

We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgo-pyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly non-Markovian) nonuniformly expanding maps hold also for their toral extensions.