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Journal of Modern Dynamics

April 2018 , Volume 12

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A quantitative Oppenheim theorem for generic ternary quadratic forms
Anish Ghosh and Dubi Kelmer
2018, 12: 1-8 doi: 10.3934/jmd.2018001 +[Abstract](1553) +[HTML](302) +[PDF](156.04KB)

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].

Values of random polynomials at integer points
Jayadev S. Athreya and Gregory A. Margulis
2018, 12: 9-16 doi: 10.3934/jmd.2018002 +[Abstract](338) +[HTML](190) +[PDF](142.38KB)

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.

Joining measures for horocycle flows on abelian covers
Wenyu Pan
2018, 12: 17-54 doi: 10.3934/jmd.2018003 +[Abstract](215) +[HTML](154) +[PDF](329.89KB)

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.

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions
Krzysztof Frączek, Ronggang Shi and Corinna Ulcigrai
2018, 12: 55-122 doi: 10.3934/jmd.2018004 +[Abstract](197) +[HTML](142) +[PDF](566.44KB)

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.

Periodic Reeb orbits on prequantization bundles
Peter Albers, Jean Gutt and Doris Hein
2018, 12: 123-150 doi: 10.3934/jmd.2018005 +[Abstract](134) +[HTML](87) +[PDF](273.4KB)

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.

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra
Aline Cerqueira, Carlos Matheus and Carlos Gustavo Moreira
2018, 12: 151-174 doi: 10.3934/jmd.2018006 +[Abstract](121) +[HTML](55) +[PDF](369.84KB)

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}.

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