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The Journal of Modern Dynamics (JMD) is dedicated to publishing research articles in active and promising areas in the theory of dynamical systems with particular emphasis on the mutual interaction between dynamics and other major areas of mathematical research, including:

  • Number theory
  • Symplectic geometry
  • Differential geometry
  • Rigidity
  • Quantum chaos
  • Teichmüller theory
  • Geometric group theory
  • Harmonic analysis on manifolds

    The journal is published by the American Institute of Mathematical Sciences (AIMS) with the support of the Center for Dynamics and Geometry at the Pennsylvania State University.

    • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
    • Publishes each article online whenever ready; the whole volume is printed in a single book at the end of the year.
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    • JMD is a publication of the American Institute of Mathematical Sciences. All rights reserved.


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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](2182) +[HTML](580) +[PDF](156.04KB)
Abstract:

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](931) +[HTML](463) +[PDF](142.38KB)
Abstract:

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](684) +[HTML](468) +[PDF](329.89KB)
Abstract:

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](782) +[HTML](495) +[PDF](566.44KB)
Abstract:

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](690) +[HTML](379) +[PDF](273.4KB)
Abstract:

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](678) +[HTML](329) +[PDF](369.84KB)
Abstract:

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}.

Rotation number of contracted rotations
Michel Laurent and Arnaldo Nogueira
2018, 12: 175-191 doi: 10.3934/jmd.2018007 +[Abstract](551) +[HTML](338) +[PDF](195.68KB)
Abstract:

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}.

Seifert manifolds admitting partially hyperbolic diffeomorphisms
Andy Hammerlindl, Rafael Potrie and Mario Shannon
2018, 12: 193-222 doi: 10.3934/jmd.2018008 +[Abstract](572) +[HTML](239) +[PDF](304.92KB)
Abstract:

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.

Continuity of Lyapunov exponents for cocycles with invariant holonomies
Lucas Backes, Aaron Brown and Clark Butler
2018, 12: 223-260 doi: 10.3934/jmd.2018009 +[Abstract](609) +[HTML](245) +[PDF](805.18KB)
Abstract:

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.

Teichmüller geodesics with $ d$-dimensional limit sets
Anna Lenzhen, Babak Modami and Kasra Rafi
2018, 12: 261-283 doi: 10.3934/jmd.2018010 +[Abstract](329) +[HTML](161) +[PDF](430.8KB)
Abstract:

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.

Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure
Ian Melbourne and Dalia Terhesiu
2018, 12: 285-313 doi: 10.3934/jmd.2018011 +[Abstract](182) +[HTML](59) +[PDF](290.51KB)
Abstract:

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.

Smooth Anosov flows: Correlation spectra and stability
Oliver Butterley and Carlangelo Liverani
2007, 1(2) : 301-322 doi: 10.3934/jmd.2007.1.301 +[Abstract](2199) +[PDF](251.1KB) Cited By(52)
Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus
Michael Brin, Dmitri Burago and Sergey Ivanov
2009, 3(1) : 1-11 doi: 10.3934/jmd.2009.3.1 +[Abstract](1980) +[PDF](143.7KB) Cited By(31)
Slow soliton interaction with delta impurities
Justin Holmer and Maciej Zworski
2007, 1(4) : 689-718 doi: 10.3934/jmd.2007.1.689 +[Abstract](1931) +[PDF](530.1KB) Cited By(29)
Fifty years of entropy in dynamics: 1958--2007
Anatole Katok
2007, 1(4) : 545-596 doi: 10.3934/jmd.2007.1.545 +[Abstract](1956) +[PDF](824.1KB) Cited By(25)
Shimura and Teichmüller curves
Martin Möller
2011, 5(1) : 1-32 doi: 10.3934/jmd.2011.5.1 +[Abstract](2125) +[PDF](645.9KB) Cited By(20)
Banach spaces for piecewise cone-hyperbolic maps
Viviane Baladi and Sébastien Gouëzel
2010, 4(1) : 91-137 doi: 10.3934/jmd.2010.4.91 +[Abstract](2000) +[PDF](552.7KB) Cited By(18)
Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups
Dmitri Burago and Sergei Ivanov
2008, 2(4) : 541-580 doi: 10.3934/jmd.2008.2.541 +[Abstract](2168) +[PDF](436.8KB) Cited By(15)
A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle
Giovanni Forni
2011, 5(2) : 355-395 doi: 10.3934/jmd.2011.5.355 +[Abstract](2224) +[PDF](435.6KB) Cited By(15)
Spectral invariants in Rabinowitz-Floer homology and global Hamiltonian perturbations
Peter Albers and Urs Frauenfelder
2010, 4(2) : 329-357 doi: 10.3934/jmd.2010.4.329 +[Abstract](1744) +[PDF](292.1KB) Cited By(15)
Schreier graphs of the Basilica group
Daniele D'angeli, Alfredo Donno, Michel Matter and Tatiana Nagnibeda
2010, 4(1) : 167-205 doi: 10.3934/jmd.2010.4.167 +[Abstract](1939) +[PDF](827.3KB) Cited By(15)
A quantitative Oppenheim theorem for generic ternary quadratic forms
Anish Ghosh and Dubi Kelmer
2018, 12: 1-8 doi: 10.3934/jmd.2018001 +[Abstract](2182) +[HTML](580) +[PDF](156.04KB) PDF Downloads(145)
Values of random polynomials at integer points
Jayadev S. Athreya and Gregory A. Margulis
2018, 12: 9-16 doi: 10.3934/jmd.2018002 +[Abstract](931) +[HTML](463) +[PDF](142.38KB) PDF Downloads(110)
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](782) +[HTML](495) +[PDF](566.44KB) PDF Downloads(83)
Joining measures for horocycle flows on abelian covers
Wenyu Pan
2018, 12: 17-54 doi: 10.3934/jmd.2018003 +[Abstract](684) +[HTML](468) +[PDF](329.89KB) PDF Downloads(74)
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](678) +[HTML](329) +[PDF](369.84KB) PDF Downloads(68)
Rotation number of contracted rotations
Michel Laurent and Arnaldo Nogueira
2018, 12: 175-191 doi: 10.3934/jmd.2018007 +[Abstract](551) +[HTML](338) +[PDF](195.68KB) PDF Downloads(62)
Periodic Reeb orbits on prequantization bundles
Peter Albers, Jean Gutt and Doris Hein
2018, 12: 123-150 doi: 10.3934/jmd.2018005 +[Abstract](690) +[HTML](379) +[PDF](273.4KB) PDF Downloads(57)
Counting closed geodesics in moduli space
Alex Eskin and Maryam Mirzakhani
2011, 5(1) : 71-105 doi: 10.3934/jmd.2011.5.71 +[Abstract](1296) +[PDF](396.8KB) PDF Downloads(49)
Continuity of Lyapunov exponents for cocycles with invariant holonomies
Lucas Backes, Aaron Brown and Clark Butler
2018, 12: 223-260 doi: 10.3934/jmd.2018009 +[Abstract](609) +[HTML](245) +[PDF](805.18KB) PDF Downloads(45)
Seifert manifolds admitting partially hyperbolic diffeomorphisms
Andy Hammerlindl, Rafael Potrie and Mario Shannon
2018, 12: 193-222 doi: 10.3934/jmd.2018008 +[Abstract](572) +[HTML](239) +[PDF](304.92KB) PDF Downloads(37)

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