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On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

Part of: Smooth dynamical systems: general theory Topological dynamics Structure and classification of infinite or finite groups Abelian groups

Published online by Cambridge University Press:  12 August 2022

André Carvalho
André Carvalho*
Affiliation:
Centre of Mathematics, University of Porto, Porto 4169-007, Portugal (andrecruzcarvalho@gmail.com)
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Abstract

We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, a concept recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion, we count the number of orbits for the action of the subgroup of fixed points (respectively periodic) points on the set of infinite fixed (respectively periodic) points. Finally, we study the dynamics of infinite points: for automorphisms and some endomorphisms, defined in a precise way, fitting a classification given by Delgado and Ventura, we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic.

Keywords

free groupfree-abelian times free groupsautomorphismsendomorphismscoarse-medianfixed pointsinfinite fixed pointsperiodic pointswandering points

MSC classification

Primary: 20E36: Automorphisms of infinite groups 37C25: Fixed points, periodic points, fixed-point index theory
Secondary: 20K25: Direct sums, direct products, etc. 20E05: Free nonabelian groups 37B25: Lyapunov functions and stability; attractors, repellers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 705 - 735
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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