Poisson suspensions and entropy for infinite transformations

Janvresse, Elise; Meyerovitch, Tom; Roy, Emmanuel; Rue, Thierry de la

doi:10.1090/s0002-9947-09-04968-x

Cited by 21 publications

(26 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us illustrate it by an example. In [9], Poisson entropy of a transformation T is defined as the Kolmogorov entropy of its Poisson suspension. Poisson entropy is shown to coincide, for quasi-finite systems (see [10] for the definition), with Krengel's and Parry's definition of entropy (see [16]), moreover it is proved a dichotomy for these systems: T , quasi-finite and ergodic, is either a remotely infinite system, or it possesses a Pinsker (σ-finite)-factor.…”

Section: Strong Disjointness and Id-disjointnessmentioning

confidence: 99%

Poisson suspensions and infinite ergodic theory

Roy

2009

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

Abstract. We investigate 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.

show abstract

Section: Strong Disjointness and Id-disjointnessmentioning

confidence: 99%

Poisson suspensions and infinite ergodic theory

Roy

2009

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…That they always do exist for free actions 3 was established by Benjamin 1 In fact, hµ(T ) can be expressed purely in terms of the size of generators: writing gµ(T ) for the cardinality of the smallest µ-generator of T , we have hµ(T ) = lim n→∞ 1 n log gµ(T n ) 2 Several notions of entropy have been suggested for conservative transformation, e.g. [16,21,13], but these generally lack many important properties present in the classical notion. 3 A free Z-action is one without periodic points.…”

Section: Background and Statement Of Resultsmentioning

confidence: 99%

Every Borel automorphism without finite invariant measures admits a two-set generator

Hochman

2018

J. Eur. Math. Soc.

View full text Add to dashboard Cite

We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator. This implies that if the entropies of invariant probability measures of a Borel system are all less than log k, then the system admits a k-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.[20], when the entropies of two such systems are equal, their entropy-maximizing measures are isomorphic. In many special cases, e.g. for mixing shifts of finite type, this isomorphism can be made continuous on a set of full measure, as in the finitary isomorphism theory of Keane and Smorodinsky [15], and even extended farther "down" to some of the "low-entropy" part of the phase, as is the almost-conjugacy theorem of Adler and Marcus [1].More recently Buzzi introduced the notion of entropy conjugacy [6], whereby in the problem above one replaces continuity by measurability in the hope of extending the isomorphism results to a larger class of systems [4,6]. One also hopes to extend the isomorphisms farther into the low-entropy part of the systems, ideally to all of the "free part" of the system, that is, to the complement of the periodic points. 5 This possibility was raised in [10], where it was partly achieved for a large family of systems on sets supporting all non-atomic invariant probability measures (but not all conservative ones). See also [7]. Isomorphisms between the entire free parts of equal-entropy strongly positively recurrent Markov shifts were constructed recently by Boyle, Buzzi and Gómez [5], using the special presentations of such subshifts. Using the arguments from [10] together with Theorem 1.1 one can give a quite general result in this direction. Corollary 1.3. Let h > 0. Then, up to Borel isomorphism, there is a unique homeomorphism T of a Polish space satisfying the following properties: (a) T acts freely, (b) every T -invariant probability measure has entropy ≤ h, and equality occurs for a unique measure which is Bernoulli, (c) T admits embedded mixing SFTs of topological entropy arbitrarily close to h.In particular, if two systems from the classes listed below have the same topological entropy, then they are isomorphic, as Borel systems, on the complements of their periodic points. The classes are: Mixing positively-recurrent countablestate shifts of finite type, mixing sofic shifts, Axiom A diffeomorphisms, intrinsically ergodic mixing shifts of quasi-finite type.It remains an open problem whether, on the complement of the periodic points, the isomorphism can be made continuous in any non-trivial cases, e.g. between equal-entropy mixing shifts of finite type which are not topologically conjugate [10, Problem 1.9].

show abstract

“…In [4] it was proved that the relative entropy of the corresponding Poisson suspensions is equal to the relative entropy of the underlying transformations, which gives a direct proof to Corollary 6.1. However, for any c > 0, there is a quasi-factor ξ which satisfies…”

Section: Poisson Suspensionsmentioning

confidence: 89%

“…By linearity of Poisson entropy (see [4]), the relative entropy h ξ (T * | S * ) is c times the relative entropy h μ (T | S).…”

Section: Poisson Suspensionsmentioning

confidence: 99%

See 1 more Smart Citation

Quasi-factors and relative entropy for infinite-measure-preserving transformations

Meyerovitch

2011

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

We extend the definition of quasi-factors for infinite-measure-preserving transformations. The existence of a system with zero Krengel entropy and a quasi-factor with positive entropy is obtained. On the other hand, relative zero-entropy for conservative systems implies relative zero-entropy of any quasi-factor with respect to its natural projection onto the factor. This extends (and is based upon) results of Glasner, Thouvenot and Weiss [6,7]. Following and extending Glasner and Weiss [8], we also prove that any conservative measure-preserving system with positive entropy in the sense of Danilenko and Rudolph [3] admits any probability-preserving system with positive entropy as a factor. Some applications and connections with Poisson-suspensions are presented.

show abstract

Poisson suspensions and entropy for infinite transformations

Cited by 21 publications

References 19 publications

Poisson suspensions and infinite ergodic theory

Poisson suspensions and infinite ergodic theory

Every Borel automorphism without finite invariant measures admits a two-set generator

Quasi-factors and relative entropy for infinite-measure-preserving transformations

Contact Info

Product

Resources

About