Recurrence in Ergodic Theory and Combinatorial Number Theory

Furstenberg, Harry

doi:10.1515/9781400855162

Cited by 1,055 publications

(601 citation statements)

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“…Using the multiple weak-mixing theorem ( [Fu81], page 86) it is easy to see that if a strongly stationary system is weak-mixing then it is Bernoulli. In [Je97] Jenvey shows that the same conclusion holds if we just assume ergodicity.…”

Section: Stationary Processesmentioning

confidence: 99%

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The structure of strongly stationary systems

Frantzikinakis

2004

J. Anal. Math.

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Abstract. Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems and rotations on nilmanifolds.

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Section: Stationary Processesmentioning

confidence: 99%

“…Let µ = µ t dλ(t) be the ergodic decomposition of µ. Letting N → ∞ and using the multiple weak mixing theorem ([Fu81], page 86) we get…”

Section: Stationary Processesmentioning

confidence: 99%

The structure of strongly stationary systems

Frantzikinakis

2004

J. Anal. Math.

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“…It is important to note that solvability of partition-regular linear systems of equations within WM sets can be shown directly (without use of Theorem 1.3.1) by use of the technique of Furstenberg and Weiss that was developed in their dynamical proof of Rado's theorem (see [8]). First of all we describe Rado's regular systems.…”

Section: Comparison With Rado's Theoremmentioning

confidence: 99%

Solvability of linear equations within weak mixing sets

Fish

2011

Isr. J. Math.

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We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that any partition-regular system of linear equations with integer coefficients is solvable in any WM set.

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“…Let (X, 0) be a dynamical system and xo G X. The point xo is said to be recurrent if, for any neighborhood V of xo, there is an integer n > 1 with 0n(x) G V. (Definition 1.1 of [6].) A subset S C {0,1,2,... } = Z+ is called syndetic if there is a finite set 7 c Z+ such that S + 7 = Z+.…”

Section: Ii2 Definitions If (X) (Yip) Are Dynamical Systems (Yipmentioning

confidence: 99%

The Ideal Structure of Certain Nonselfadjoint Operator Algebras

Peters¹

1988

Transactions of the American Mathematical Society

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ABSTRACT. Let (X,4>) be a locally compact dynamical system, and Z+ x$ Ccj(X) the norm-closed subalgebra of the crossed product Z x^ Cq(X) generated by the nonnegative powers of in case 0 is a homeomorphism.If is just a continuous map, Z+ x^ Co(X) can still be defined by a crossed product type construction.The ideal structure of these algebras is determined in case acts freely. A class of strictly transitive Banach modules is described, indicating that for the nonselfadjoint operator algebras considered here, not all irreducible representations are on Hubert space. Finally in a special case, the family of all invariant maximal right ideals is given.

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Recurrence in Ergodic Theory and Combinatorial Number Theory

Cited by 1,055 publications

References 0 publications

The structure of strongly stationary systems

The structure of strongly stationary systems

Solvability of linear equations within weak mixing sets

The Ideal Structure of Certain Nonselfadjoint Operator Algebras

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