# Reeb dynamics inspired by Katok's example in Finsler geometry

## TL;DR

This paper explores Reeb flows with finitely many periodic orbits inspired by Katok's Finsler metrics, providing new contact-geometric interpretations and constructions in higher dimensions that contrast with known results in three dimensions.

## Contribution

It introduces novel methods to produce Reeb flows with arbitrarily many periodic orbits in higher dimensions using Hamiltonian circle actions and surgery techniques.

## Key findings

- Contact-geometric interpretation of magnetic flows on the 2-sphere.
- Construction of contact manifolds with arbitrarily many periodic Reeb orbits in dimensions ≥5.
- Reproduction of Hamiltonian flows with any number of periodic orbits in dimensions ≥3.

## Abstract

Inspired by Katok's examples of Finsler metrics with a small number of closed geodesics, we present two results on Reeb flows with finitely many periodic orbits. The first result is concerned with a contact-geometric description of magnetic flows on the 2-sphere found recently by Benedetti. We give a simple interpretation of that work in terms of a quaternionic symmetry. In the second part, we use Hamiltonian circle actions on symplectic manifolds to produce compact, connected contact manifolds in dimension at least five with arbitrarily large numbers of periodic Reeb orbits. This contrasts sharply with recent work by Cristofaro-Gardiner, Hutchings and Pomerleano on Reeb flows in dimension three. With the help of Hamiltonian plugs and a surgery construction due to Laudenbach we reprove a result of Cieliebak: one can produce Hamiltonian flows in dimension at least five with any number of periodic orbits; in dimension three, with any number greater than one.