ISSN:
1930-5311
eISSN:
1930-532X
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Journal of Modern Dynamics
January 2014 , Volume 8 , Issue 1
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The 2013 Michael Brin Prize in Dynamical Systems
(Brin Prize article)
2014, 8(1): i-ii
doi: 10.3934/jmd.2014.8.1i
+[Abstract](3945)
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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.
For more information please click the “Full Text” above.
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.
For more information please click the “Full Text” above.
On the work of Sarig on countable Markov chains and thermodynamic formalism
(Brin Prize article)
2014, 8(1): 1-14
doi: 10.3934/jmd.2014.8.1
+[Abstract](4024)
+[PDF](180.7KB)
Abstract:
The paper is a nontechnical survey and is aimed to illustrate Sarig'sprofound contributions to statistical physics and in particular,thermodynamic formalism for countable Markov shifts. I will discusssome of Sarig's work on characterization of existence of Gibbsmeasures, existence and uniqueness of equilibrium states as well asphase transitions for Markov shifts on a countable set of states.
The paper is a nontechnical survey and is aimed to illustrate Sarig'sprofound contributions to statistical physics and in particular,thermodynamic formalism for countable Markov shifts. I will discusssome of Sarig's work on characterization of existence of Gibbsmeasures, existence and uniqueness of equilibrium states as well asphase transitions for Markov shifts on a countable set of states.
On Omri Sarig's work on the dynamics on surfaces
(Brin Prize article)
2014, 8(1): 15-24
doi: 10.3934/jmd.2014.8.15
+[Abstract](3135)
+[PDF](176.5KB)
Abstract:
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2014, 8(1): 25-59
doi: 10.3934/jmd.2014.8.25
+[Abstract](2998)
+[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.
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.
2014, 8(1): 61-73
doi: 10.3934/jmd.2014.8.61
+[Abstract](3110)
+[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.
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.
2014, 8(1): 75-91
doi: 10.3934/jmd.2014.8.75
+[Abstract](3317)
+[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.
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.
2014, 8(1): 93-107
doi: 10.3934/jmd.2014.8.93
+[Abstract](3157)
+[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.
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.
2014, 8(1): 109-132
doi: 10.3934/jmd.2014.8.109
+[Abstract](3757)
+[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.
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.
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