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Generic points in the cartesian powers of the Morse dynamical system

Emmanuel Lesigne;  Anthony Quas;  Máté Wierdl — 2003

Bulletin de la Société Mathématique de France

The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.

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.

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