On Monotonic Fixed-Point Free Bijections on Subgroups of $\mathbb{R}$

Abstract

We show that for any continuous monotonic fixed-point free automorphism $f$ on a $\sigma$-compact subgroup $G\subset \mathbb{R}$ there exists a binary operation ${+}_f$ such that $⟨G,{+}_f⟩$ is a topological group topologically isomorphic to $⟨G,+⟩$ and $f$ is a shift with respect to ${+}_f$. We then show that monotonicity cannot be replaced by the property of being periodic-point free. We explore a few routes leading to generalizations and counterexamples.