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Journal of Modern Dynamics

2014 , Volume 8 , Issue 1

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The 2013 Michael Brin Prize in Dynamical Systems
The Editors
2014, 8(1): i-ii doi: 10.3934/jmd.2014.8.1i +[Abstract](319) +[PDF](9238.9KB)
Abstract:
Professor Michael Brin of the University of Maryland endowed an international prize for outstanding work in the theory of dynamical systems and related areas. The prize is given biennially for specific mathematical achievements that appear as a single publication or a series thereof in refereed journals, proceedings or monographs.

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On the work of Sarig on countable Markov chains and thermodynamic formalism
Yakov Pesin
2014, 8(1): 1-14 doi: 10.3934/jmd.2014.8.1 +[Abstract](495) +[PDF](180.7KB)
Abstract:
The paper is a nontechnical survey and is aimed to illustrate Sarig's profound contributions to statistical physics and in particular, thermodynamic formalism for countable Markov shifts. I will discuss some of Sarig's work on characterization of existence of Gibbs measures, existence and uniqueness of equilibrium states as well as phase transitions for Markov shifts on a countable set of states.
On Omri Sarig's work on the dynamics on surfaces
François Ledrappier
2014, 8(1): 15-24 doi: 10.3934/jmd.2014.8.15 +[Abstract](356) +[PDF](176.5KB)
Abstract:
N/A
Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows
Alexander Gorodnik and Frédéric Paulin
2014, 8(1): 25-59 doi: 10.3934/jmd.2014.8.25 +[Abstract](416) +[PDF](1361.8KB)
Abstract:
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral points on sections of quadrics. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.
Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum
Julien Grivaux and Pascal Hubert
2014, 8(1): 61-73 doi: 10.3934/jmd.2014.8.61 +[Abstract](360) +[PDF](193.5KB)
Abstract:
We construct explicit closed $\mathrm{GL}(2; \mathbb{R})$-invariant loci in strata of meromorphic quadratic differentials of arbitrarily large dimension with fully degenerate Lyapunov spectrum. This answers a question of Forni-Matheus-Zorich.
Topological entropy of minimal geodesics and volume growth on surfaces
Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou and Jan Philipp Schröder
2014, 8(1): 75-91 doi: 10.3934/jmd.2014.8.75 +[Abstract](465) +[PDF](203.7KB)
Abstract:
Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type, i.e $M$ is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of $(M,g)$ generalizing work of Freire and Mañé.
Minimal yet measurable foliations
Gabriel Ponce, Ali Tahzibi and Régis Varão
2014, 8(1): 93-107 doi: 10.3934/jmd.2014.8.93 +[Abstract](443) +[PDF](197.7KB)
Abstract:
In this paper we mainly address the problem of disintegration of Lebesgue measure along the central foliation of volume-preserving diffeomorphisms isotopic to hyperbolic automorphisms of 3-torus. We prove that atomic disintegration of the Lebesgue measure (ergodic case) along the central foliation has the peculiarity of being mono-atomic (one atom per leaf). This implies the measurability of the central foliation. As a corollary we provide open and nonempty subset of partially hyperbolic diffeomorphisms with minimal yet measurable central foliation.
Pseudo-integrable billiards and arithmetic dynamics
Vladimir Dragović and Milena Radnović
2014, 8(1): 109-132 doi: 10.3934/jmd.2014.8.109 +[Abstract](487) +[PDF](783.0KB)
Abstract:
We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behavior different from Liouville--Arnold's Theorem. Its analog is derived from the Maier Theorem on measured foliations. The billiard flow generates a measurable foliation defined by a closed 1-form $w$. Using the closed form, a transformation of the given billiard table to a rectangular cylinder is constructed and a trajectory equivalence between corresponding billiards has been established. A local version of Poncelet Theorem is formulated and necessary algebro-geometric conditions for periodicity are presented. It is proved that the dynamics depends on arithmetic of rotation numbers, but not on geometry of a given confocal pencil of conics.
Erratum: Billiards in nearly isosceles triangles
W. Patrick Hooper and Richard Evan Schwartz
2014, 8(1): 133-137 doi: 10.3934/jmd.2014.8.133 +[Abstract](372) +[PDF](430.4KB)
Abstract:
N/A

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