On monotonic bijections on subgroups of R

Raushan Buzyakova


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


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

Subject classification

06F15; 54H11; 26A48

Full Text:



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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