Hofer's norm and disk translations in an annulus

TL;DR

This paper establishes a lower bound on the Hofer's norm for Hamiltonian diffeomorphisms preserving a disk in an annulus with a given translation number, and provides examples of more efficient Hamiltonians.

Contribution

It introduces a lower bound on Hofer's norm related to translation number and constructs more efficient Hamiltonians than simple rotations.

Findings

Hofer's norm is bounded below by a constant times the translation number.

Examples of Hamiltonians more efficient than rotations are provided.

The results connect disk translations in annuli with Hofer's geometry.

Abstract

Let D be a non-displaceable disk in an annulus A. Suppose that g is a compactly supported Hamiltonian which preserves D with translation number n. We show that Hofer's norm |g| is bounded from below by cn for a certain constant c. We also give example of such Hamiltonian which is more efficient than the obvious rotation of D.