Real recurrence sets

Thesis

Classical theorems such as the Poincaré recurrence theorem, the van der Corput theorem about equidistribution of sequences or the F. and M. Riesz theorem about measures on the torus show properties of the set ℕ. However, In particular the Poincaré recurrence easily allows to extend the recurrence result to sets such as mℕ for any m ∈ ℕ.

This motivates the notion of recurrence sets, i.e. sets Ɗ ⊆ ℕ (or Ɗ ⊆ ℤ) which are "strong" enough to force certain recurrence properties, and a thorough study of these sets and their relations with each other has been undertaken since the late 1970s.

This thesis deals with real recurrence sets Ɗ ⊆ ℝ. Our first result shows that integer properties and most associated implications can be transferred to the real setting and allow a similar treatment.

For a set Ɗ⊆ ℤ, the integer and real recurrence property coincide for many properties. This gives non-trivial recurrence examples from the integer theory, but also yields some counterexamples showing that some recurrence properties are distinct.

Using continuity and appropriate product systems, we show that we can reduce recurrence sets such as Poincaré or operator recurrence sets, in particular, every such recurrence set has a countable subset ^~_D⊆   ^∪_|n|>N([sn-ε,sn+ε] ∩ Ɗ) for arbitrary small ε > 0 and arbitrarily large N ∈ ℕ having the same recurrence property.

We finally indicate how to further extend this topic by discussing topological dynamical systems, a quantitative analysis of recurrence sets and the use of locally compact abelian groups instead of ℤ and ℝ.

Authors: Konrad, L

• Publication date: 2015
• Type of award: MSc by Research
• Level of award: Masters
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: Recurrence