scholarly journals Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure

2018 ◽  
Vol 12 (1) ◽  
pp. 285-313
Author(s):  
Ian Melbourne ◽  
◽  
Dalia Terhesiu ◽  
Keyword(s):  
Dynamical Systems ◽  
Mixing Properties ◽  
Infinite Measure

scholarly journals Poisson suspensions and infinite ergodic theory

2009 ◽  
Vol 29 (2) ◽  
pp. 667-683 ◽  
Author(s):  
EMMANUEL ROY
Keyword(s):  
Dynamical Systems ◽  
Spectral Theory ◽  
Ergodic Theory ◽  
Mixing Properties ◽  
New Approach ◽  
Infinite Measure ◽  
Infinite Ergodic Theory

AbstractWe investigate the ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure-preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite-measure ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems.

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.

Sequence Entropy and Mild Mixing

1992 ◽  
Vol 44 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Qing Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Infinite Subset ◽  
Mixing Properties ◽  
Sequence Entropy ◽  
Number Of Authors

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.

Improved mixing rates for infinite measure-preserving systems

2013 ◽  
Vol 35 (2) ◽  
pp. 585-614 ◽  
Author(s):  
DALIA TERHESIU
Keyword(s):  
Dynamical Systems ◽  
Simple Proof ◽  
Renewal Theory ◽  
New Technique ◽  
First Order ◽  
Return Map ◽  
Infinite Measure ◽  
Mixing Rates ◽  
A New Technique ◽  
Invent Math

AbstractIn this work, we introduce a new technique for operator renewal sequences associated with dynamical systems preserving an infinite measure that improves the results on mixing rates obtained by Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61–110]. Also, this technique allows us to offer a very simple proof of the key result of Melbourne and Terhesiu that provides first-order asymptotics of operator renewal sequences associated with dynamical systems with infinite measure. Moreover, combining techniques used in this work with techniques used by Melbourne and Terhesiu, we obtain first-order asymptotics of operator renewal sequences under some relaxed assumption on the first return map.

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