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Ergodic Theory: Recurrence

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Ergodic Theory

Part of the book series: Encyclopedia of Complexity and Systems Science Series ((ECSSS))

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  • R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, © Springer Science+Business Media LLC 2021

Glossary

Almost every, essentially:

Given a Lebesgue measure space (X, ℬ, μ), a property P(x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x: P (x) holds} has zero measure. Two sets A, B ∈ ℬ are essentially disjoint if μ(A ∩ B) = 0.

Conservative system:

Is an infinite measure preserving system such that for no set A ∈ ℬ with positive measure are A, T−1A, T−2A, … pairwise essentially disjoint.

(cn)-Conservative system:

If (cn)n∈ℕ is a decreasing sequence of positive real numbers, a conservative ergodic measure preserving transformation T is (cn)-conservative if for some nonnegative function f ∈ L1(μ), \( {\sum}_{n=1}^{\infty }{c}_n\, f\left({T}^nx\right)=\infty \) a.e.

Doubling map:

If \( \mathbb{T} \) is the interval [0, 1] with its endpoints identified and addition performed modulo 1, the (non-invertible) transformation T: \( \mathbb{T} \) → \( \mathbb{T} \), defined by Tx = 2xmod 1, preserves Lebesgue measure, hence induces a measure...

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Authors and Affiliations

  1. Department of Mathematics, University of Crete, Heraklion, Greece

    Nikos Frantzikinakis

  2. Department of Mathematics, University of Memphis, Memphis, TN, USA

    Randall McCutcheon

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  1. Nikos Frantzikinakis
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  2. Randall McCutcheon
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Correspondence to Nikos Frantzikinakis .

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  1. Mathematics, Williams College, Williamstown, MA, USA

    Cesar E. Silva

  2. B.Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv, Ukraine

    Alexandre I. Danilenko

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Frantzikinakis, N., McCutcheon, R. (2023). Ergodic Theory: Recurrence. In: Silva, C.E., Danilenko, A.I. (eds) Ergodic Theory. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-2388-6_184

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