Ljapunov Characteristic Exponents and Ergodic Properties of Smooth Dynamical Systems with an Invariant Measure

2020 ◽  
pp. 512-515
Author(s):  
Ja. B. Pesin
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Characteristic Exponents ◽  
Ergodic Properties

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.

Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties

1992 ◽  
Vol 12 (1) ◽  
pp. 123-151 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Invariant Measure ◽  
Topological Properties ◽  
Smooth Measure ◽  
Ergodic Properties ◽  
Special Invariant ◽  
Hyperbolic Attractors ◽  
Hyperbolic Behavior

AbstractWe introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

Mixing for invertible dynamical systems with infinite measure

2015 ◽  
Vol 15 (02) ◽  
pp. 1550012 ◽  
Author(s):  
Ian Melbourne
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Large Class ◽  
Renewal Theory ◽  
Additional Structure ◽  
Infinite Measure ◽  
Higher Order Asymptotics ◽  
Mixing Rates ◽  
Noninvertible Maps ◽  
Invent Math

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math.189 (2012) 61–110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure. Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates and higher order asymptotics.

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

scholarly journals Logical entropy of quantum dynamical systems

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Abolfazl Ebrahimzadeh
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Quantum Logic ◽  
Quantum Dynamical ◽  
Quantum Dynamical System ◽  
Ergodic Properties ◽  
Quantum Dynamical Systems ◽  
Logical Entropy

AbstractThis paper introduces the concepts of logical entropy and conditional logical entropy of hnite partitions on a quantum logic. Some of their ergodic properties are presented. Also logical entropy of a quantum dynamical system is dehned and ergodic properties of dynamical systems on a quantum logic are investigated. Finally, the version of Kolmogorov-Sinai theorem is proved.

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