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Morphisms of groupoid actions and recurrence. (English) Zbl 1523.22008

In a recent paper [F. Flores and M. Măntoiu, Groups Geom. Dyn. 16, No. 3, 1005–1047 (2022; Zbl 1526.22005)] the authors studied dynamical aspects of continuous groupoid actions, including notions such as transitivity, minimality, (non) wandering, recurrence, and (almost) periodicity. They also investigated how epimorphisms preserve these properties, where for a topological groupoid acting on two topological spaces, an epimorphism consists of an equivariant continuous surjection between the underlying topological spaces. In this setting, the spaces change but the acting groupoid is the same.
The main objective of the paper under review is to settle the case where acting groupoids are not the same. In this more general case, a natural candidate for a morphism (epimorphism) is a pair consisting of a continuous functor between the acting groupoids – considered as small categories – and an equivariant continuous map (surjection) between the underlying topological spaces, with natural compatibility conditions. This is defined and briefly discussed in the paper, but soon extended to the notion of generalized vague morphisms. This extended notion is adapted from [R. Meyer and C. Zhu, Theory Appl. Categ. 30, 1906–1998 (2015; Zbl 1330.18005)], and instead of functors between the acting groupoids uses morphisms between the corresponding pullbacks, which are then paired with a continuous function between underlying topological spaces with suitable compatiblity conditions. The authors also suggest another notion for a morphism, which uses the so called algebraic morphisms between groupoids, essentially appeared for the first time in [S. Zakrzewski, Commun. Math. Phys. 134, No. 2, 347–370 (1990; Zbl 0708.58030); Commun. Math. Phys. 134, No. 2, 371–395 (1990; Zbl 0708.58031)]. The idea is that instead of using a map between the groupoids, one could use a continuous action of the first groupoid on the second – regraded as a topological space – commuting with the action of the second groupoid on itself. As observed in [A. Buss et al., Semigroup Forum 85, No. 2, 227–243 (2012; Zbl 1261.22003)], this is the same as considering functors between the categories of actions, keeping the underlying space unchanged.

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
58H05 Pseudogroups and differentiable groupoids
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References:

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