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A non-autonomous Hamiltonian diffeomorphism with roots of all orders

Nicolas Grunder, Baptiste Serraille

Abstract

We present a way of constructing non-autonomous Hamiltonian diffeomorphisms with roots of all orders by adapting the Anosov-Katok construction. This answers a question by Kathryn Mann and Egor Shelukin. Additionally, we construct an action of the rationals by diffeomorphism on any manifold that is not $C^0$-continuous with respect to the Euclidean topology on $\mathbb Q$.

A non-autonomous Hamiltonian diffeomorphism with roots of all orders

Abstract

We present a way of constructing non-autonomous Hamiltonian diffeomorphisms with roots of all orders by adapting the Anosov-Katok construction. This answers a question by Kathryn Mann and Egor Shelukin. Additionally, we construct an action of the rationals by diffeomorphism on any manifold that is not -continuous with respect to the Euclidean topology on .

Paper Structure

This paper contains 13 sections, 18 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Main result and motivation
  3. Organization of the paper
  4. Acknowledgments
  5. Proof of Proposition \ref{['prop: non-aut with roots']}
  6. Preliminaries of symplectic geometry
  7. Setup of the Anosov-Katok construction
  8. Scheme of the construction
  9. Achieving dense orbits
  10. Proof of Theorem \ref{['thm: non-continuous']}
  11. Constructing exotic rationals in $\mathrm{Diff}(M)$
  12. Relation with the $\gamma$-topology and Hofer-topology
  13. The special case of manifolds with a Hamiltonian circle action

Key Result

Theorem 1

For any symplectic manifold $(M,\omega)$ there exists an injective group homomorphism such that $\Phi(\mathbb Q)\cap \mathrm{Auto}(M,\omega) = \{id\}.$

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 1.1
  • Proposition 1.2
  • proof : Proof of Theorem \ref{['thm: Q to Ham']}
  • Theorem 2
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • ...and 21 more