On monotonic bijections on subgroups of R

Raushan Buzyakova

Abstract

We show that for any continuous monotonic  bijection $f$ on a $\sigma$-compact subgroup  $G\subset \mathbb R$ there exists a binary operation $+_f$ such that $\langle G, +_f\rangle$  is a topological group topologically isomorphic to $\langle G, +\rangle$ and $f$ is a shift with respect to $+_f$. We then show that monotonicity cannot be replaced by a periodic-point free continuous bijections. We explore a few routes leading to generalizations and counterexamples

Keywords

ordered group; topological group; homeomorphism, shift; monotonic function; fixed point; periodic point.

Subject classification

06F15; 54H11; 26A48.

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References

P. Alexandroff and P. Urysohn, Uber nuldimensionale Punktmengen, Math Ann. 98 (1928), 89-106.

(http://dx.doi.org/10.1007/BF01451582)

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W. Sierpinski, Sur une propriete topologique des ensembles denombrablesdense en soi, Fund. Math. 1 (1920), 11-16.

J. van Mill, The Infinite-Dimensional Topology of Function Spaces, Elsevier, 2001

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1. A glance into the anatomy of monotonic maps
Raushan Buzyakova
Applied General Topology  vol: 22  issue: 1  first page: 1  year: 2021  
doi: 10.4995/agt.2021.12291



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt