Avalanche dynamics

2016 ◽  
Vol 16 (02) ◽  
pp. 1660005 ◽  
Author(s):  
Manfred Denker ◽  
Anna Levina
Keyword(s):  
Invariant Measure ◽  
Group Actions ◽  
Neural Dynamics ◽  
Open Problems ◽  
Random Dynamics ◽  
Skew Products ◽  
Avalanche Dynamics ◽  
Finite Invariant Measure

The avalanche transformation as a model for avalanches in neural dynamics was introduced in [8] in 2008. Here we discuss this transformation in terms of group actions, random dynamics and skew products with a finite invariant measure. The results are based on [8]. Some open problems are mentioned.

On Finite Invariant Measure for Semigroups of Operators

1971 ◽  
Vol 14 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Usha Sachdevao
Keyword(s):  
Invariant Measure ◽  
Semigroups Of Operators ◽  
Amenable Semigroup ◽  
Linear Contraction ◽  
Finite Invariant Measure

Let Σ be a left amenable semigroup, and let {Tσ: σ ∊ Σ} be a representation of Σ as a semigroup of positive linear contraction operators on L1(X, 𝓐, p). This paper is devoted to the study of existence of a finite equivalent invariant measure for such semigroups of operators.

scholarly journals Orbit equivalence and rigidity of ergodic actions of Lie groups

1981 ◽  
Vol 1 (2) ◽  
pp. 237-253 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Invariant Measure ◽  
Euclidean Space ◽  
Lie Groups ◽  
Homogeneous Spaces ◽  
The Other ◽  
Rigidity Theorem ◽  
Simple Groups ◽  
Orbit Equivalence ◽  
Finite Invariant Measure ◽  
Lattice Subgroups

AbstractThe rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

scholarly journals Measure rigidity for random dynamics on surfaces and related skew products

2017 ◽  
Vol 30 (4) ◽  
pp. 1055-1132 ◽  
Author(s):  
Aaron Brown ◽  
Federico Rodriguez Hertz
Keyword(s):  
Random Dynamics ◽  
Skew Products ◽  
Measure Rigidity

Ergodic elements for actions of Lie groups

1996 ◽  
Vol 16 (4) ◽  
pp. 703-717
Author(s):  
K. Robert Gutschera
Keyword(s):  
Invariant Measure ◽  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Group Structure ◽  
Structure Theory ◽  
Ergodic Action ◽  
Finite Invariant Measure ◽  
Connected Lie Group

AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

scholarly journals Discrete spectrum for amenable group actions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tao Yu ◽  
Guohua Zhang ◽  
Ruifeng Zhang
Keyword(s):  
Invariant Measure ◽  
Discrete Spectrum ◽  
Invariant Measures ◽  
Hamming Distance ◽  
Amenable Group ◽  
Group Actions ◽  
The Mean

<p style='text-indent:20px;'>In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions.</p><p style='text-indent:20px;'>We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.</p>

scholarly journals Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés

1989 ◽  
Vol 9 (3) ◽  
pp. 433-453 ◽  
Author(s):  
Y. Guivarc'h
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Manifolds ◽  
Constant Negative Curvature ◽  
Skew Products ◽  
Ergodic Properties ◽  
Anosov Systems

AbstractWe study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to theKproperty of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

scholarly journals Existence and Non-existence of a Finite Invariant Measure

2012 ◽  
Vol 35 (2) ◽  
pp. 339-358
Author(s):  
Stanley EIGEN ◽  
Arshag HAJIAN ◽  
Yuji ITO ◽  
Vidhu S. PRASAD
Keyword(s):  
Invariant Measure ◽  
Finite Invariant Measure
Sign in / Sign up

Export Citation Format

Share Document