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On nilspace systems and their morphisms

Part of: Topological dynamics Connections with other structures, applications

Published online by Cambridge University Press:  10 April 2019

PABLO CANDELA ,
DIEGO GONZÁLEZ-SÁNCHEZ  and
BALÁZS SZEGEDY
PABLO CANDELA
Affiliation:
Universidad Autónoma de Madrid and ICMAT, Ciudad Universitaria de Cantoblanco, Madrid28049, Spain email [email protected]
DIEGO GONZÁLEZ-SÁNCHEZ
Affiliation:
Universidad Autónoma de Madrid and ICMAT, Ciudad Universitaria de Cantoblanco, Madrid28049, Spain email [email protected]
BALÁZS SZEGEDY
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13–15, Budapest1053, Hungary email [email protected]
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Abstract

A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace $\text{X}$ equipped with a group of nilspace translations acting on $\text{X}$. Nilspace systems appear in different guises in several recent works and this motivates the study of these systems per se as well as their relation to more classical types of systems. In this paper we study morphisms of nilspace systems, i.e., nilspace morphisms with the additional property of being consistent with the actions of the given translations. A nilspace morphism does not necessarily have this property, but one of our main results shows that it factors through some other morphism which does have the property. As an application we obtain a strengthening of the inverse limit theorem for compact nilspaces, valid for nilspace systems. This is used in work of the first- and third-named authors to generalize the celebrated structure theorem of Host and Kra on characteristic factors.

Keywords

compact nilspacesnilspace systemsnilsystemsmorphisms

MSC classification

Primary: 37B99: None of the above, but in this section
Secondary: 54H20: Topological dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 3015 - 3029
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Copyright
© Cambridge University Press, 2019

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