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There exist no contact Anosov diffeomorphisms

Masayuki Asaoka, Yoshihiko Mitsumatsu

TL;DR

This work proves there are no contact Anosov diffeomorphisms on closed odd-dimensional manifolds. For a closed manifold $M^{2m+1}$ and a $C^2$-Anosov diffeomorphism $f$, no invariant contact structure $\xi$ can be preserved by $f$. The argument combines an invariant-distribution propagation along stable leaves, a splitting $E^s= auigoplus(oldsymbol{ heta}ig|_{E^s})$, and a pullback estimate that forces a global scaling on a contact form, leading to a volume contradiction and, ultimately, to a contradiction with the nondegeneracy of $doldsymbol{ heta}$ on $oldsymbol{ heta}$. This rules out the existence of contact-Anosov diffeomorphisms and has implications for related questions about approximations by contact diffeomorphisms and convex hypersurface realizability in higher dimensions.

Abstract

For any Anosov diffeomorphims on a closed odd dimensional manifold, there exists no invariant contact structure.

There exist no contact Anosov diffeomorphisms

TL;DR

This work proves there are no contact Anosov diffeomorphisms on closed odd-dimensional manifolds. For a closed manifold and a -Anosov diffeomorphism , no invariant contact structure can be preserved by . The argument combines an invariant-distribution propagation along stable leaves, a splitting , and a pullback estimate that forces a global scaling on a contact form, leading to a volume contradiction and, ultimately, to a contradiction with the nondegeneracy of on . This rules out the existence of contact-Anosov diffeomorphisms and has implications for related questions about approximations by contact diffeomorphisms and convex hypersurface realizability in higher dimensions.

Abstract

For any Anosov diffeomorphims on a closed odd dimensional manifold, there exists no invariant contact structure.
Paper Structure (2 sections, 2 theorems, 10 equations)

This paper contains 2 sections, 2 theorems, 10 equations.

Table of Contents

  1. Result
  2. Proof

Key Result

Theorem 1.1

Let $M$ be an odd dimensional closed manifold and $f$ a $C^2$ Anosov diffeomorphism on $M$. Then, $f$ preserves no contact structures on $M$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof