Nilsequences and a structure theorem for topological dynamical systems
By B. Host, Bryna Kra, Alejandro Maass. Advances in Mathematics, 2009
96 citations recorded by the source corpus.
Abstract We characterize inverse limits of nilsystems in topological dynamics, via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems. We provide two applications of the structure. The first is to nilsequences, which have played an important role in recent developments in ergodic theory and additive combinatorics; we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally, meaning on finite intervals. The second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system. We show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bb996c74b657a31b5af08fed610f8b2497cc8784
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In Discrete Mathematics and Combinatorics
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Citations in the corpus
- Infinite-step nilsystems, independence and complexity (2011, background): “2 To show the next lemma, we need some results from [16].”
- On automorphism groups of low complexity subshifts (2015, methodology): “ace Zd(X) = X/RP[d](X). Moreover, the natural projection πd : (X,T) → (Zd(X),Td) defines a topological factor map. The following theorem describes the topological structure of (Zd(X),Td). Theorem 5.2 ([24]). Let (X,T) be a minimal topological dynamical system. For each integer d≥ 1, (Zd(X),Td) is top”
- Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence (2010, methodology): “First recall that (Q,G) is a minimal system, which is mentioned in [22].”