Homogeneous Dynamics in the Plane

1994 ◽  
pp. 163-209
Author(s):  
Bjarne S. Jensen
Keyword(s):  
Homogeneous Dynamics

scholarly journals Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

2017 ◽  
Vol 38 (7) ◽  
pp. 2780-2800 ◽  
Author(s):  
RODOLPHE RICHARD ◽  
NIMISH A. SHAH
Keyword(s):  
Number Theory ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Reductive Groups ◽  
Natural Group ◽  
Homogeneous Dynamics ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Unipotent Flows ◽  
Linearization Techniques

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

scholarly journals Homogeneous dynamics in a vibrated granular monolayer

2019 ◽  
Vol 2019 (9) ◽  
pp. 093205 ◽  
Author(s):  
P Maynar ◽  
M I García de Soria ◽  
J Javier Brey
Keyword(s):  
Homogeneous Dynamics

scholarly journals Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

2019 ◽  
Author(s):  
Corina Ciobotaru ◽  
Vladimir Finkelshtein ◽  
Cagri Sert
Keyword(s):  
Large Class ◽  
Closed Subgroup ◽  
Discrete Subgroup ◽  
Probability Measures ◽  
Homogeneous Dynamics ◽  
Cocompact Lattice ◽  
Geometrically Finite ◽  
Measure Rigidity ◽  
Groups Acting On Trees

Abstract We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $G$, we give a classification of the probability measures on $G/\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

Homogeneous Dynamics within Inhomogeneous Environment in Semicrystalline Polymers

2011 ◽  
Vol 44 (20) ◽  
pp. 8124-8128 ◽  
Author(s):  
Alejandro Sanz ◽  
Aurora Nogales ◽  
Tiberio A. Ezquerra ◽  
Wolfgang Häussler ◽  
Michelina Soccio ◽  
...  
Keyword(s):  
Semicrystalline Polymers ◽  
Homogeneous Dynamics

scholarly journals Expanding Cone and Applications to Homogeneous Dynamics

2019 ◽  
Author(s):  
Ronggang Shi
Keyword(s):  
Normal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Error Rate ◽  
Spectral Gap ◽  
General Setting ◽  
Semisimple Lie Group ◽  
Homogeneous Dynamics ◽  
Maximal Split ◽  
Split Torus

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

Measures of intermediate entropies and homogeneous dynamics

Nonlinearity ◽  
2017 ◽  
Vol 30 (9) ◽  
pp. 3349-3361 ◽  
Author(s):  
Lifan Guan ◽  
Peng Sun ◽  
Weisheng Wu
Keyword(s):  
Homogeneous Dynamics

scholarly journals Dimension estimates for sets of uniformly badly approximable systems of linear forms

2015 ◽  
Vol 11 (07) ◽  
pp. 2037-2054 ◽  
Author(s):  
Ryan Broderick ◽  
Dmitry Kleinbock
Keyword(s):  
Hausdorff Dimension ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
Upper And Lower Bounds ◽  
Homogeneous Dynamics ◽  
Linear Forms ◽  
One Dimensional ◽  
Approximation Constant ◽  
The One ◽  
Badly Approximable

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

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