On recurrence for ℤ^d-Weyl systems

2024/02/26 by Sebastián Donoso, Donoso, Sebastián, Felipe Hernández +3
Mathematics · Computer Science · #Graph theory and applications #Markov Chains and Monte Carlo Methods #Cellular Automata and Applications

paper · pdf · doi:10.48550/arxiv.2402.16838

Abstract

We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. In the case of ℤ^d-actions we develop new techniques to analyze Bohr recurrence sets. These techniques include finding and manipulating correlations between the coordinates of the set of recurrence. Using this, we show that Bohr recurrence sets are recurrence sets for ℤ^d-Weyl systems. This family encompasses, for example, all ℤ^d-affine nilsystems. To our knowledge, this is the first result towards a positive answer to Katznelson question in the case of ℤ^d-actions.

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