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

TL;DR

This paper demonstrates that continuous monotonic fixed-point free automorphisms on certain subgroups of real numbers can be represented as shifts in a suitably defined topological group, with implications for generalizations and limitations.

Contribution

It introduces a method to represent such automorphisms as shifts via a new binary operation, expanding understanding of automorphisms on subgroups of real numbers.

Findings

Existence of a binary operation making the automorphism a shift

Monotonicity is essential; periodic-point free property is insufficient

Exploration of potential generalizations and counterexamples

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 $\langle G, +_f\rangle$ 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 the property of being periodic-point free. We explore a few routes leading to generalizations and counterexamples.