On monotonic bijections on subgroups of R
Submitted: 2015-09-12
|Accepted: 2016-06-01
|Published: 2016-10-03
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Keywords:
ordered group, topological group, homeomorphism, shift, monotonic function, fixed point, periodic point
Supporting agencies:
This research was not funded
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
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References:
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