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Sets of k -recurrence but not ( k + 1 ) -recurrence

Nikos Frantzikinakis, Emmanuel Lesigne, Máté Wierdl (2006)

Annales de l’institut Fourier

For every k ∈ ℕ , we produce a set of integers which is k -recurrent but not ( k + 1 ) -recurrent. This extends a result of Furstenberg who produced a 1 -recurrent set which is not 2 -recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

Tim Austin (2016)

Analysis and Geometry in Metric Spaces

Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain...

Théorème de Kurosh pour les relations d’équivalence boréliennes

Aurélien Alvarez (2010)

Annales de l’institut Fourier

En théorie des groupes, le théorème de Kurosh est un résultat de structure concernant les sous-groupes d’un produit libre de groupes. Le théorème principal de cet article est un résultat analogue dans le cadre des relations d’équivalence boréliennes à classes dénombrables, que nous démontrons en développant une théorie de Bass-Serre dans ce cadre particulier.

Weak Closure Theorem fails for ℤ²-actions

T. Downarowicz, J. Kwiatkowski (2002)

Studia Mathematica

We construct an example of a Morse ℤ²-action which has rank one and whose centralizer contains elements which cannot be weakly approximated by the transformations of the action.

Weakly mixing rank-one transformations conjugate to their squares

Alexandre I. Danilenko (2008)

Studia Mathematica

Utilizing the cut-and-stack techniques we construct explicitly a weakly mixing rigid rank-one transformation T which is conjugate to T². Moreover, it is proved that for each odd q, there is such a T commuting with a transformation of order q. For any n, we show the existence of a weakly mixing T conjugate to T² and whose rank is finite and greater than n.

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