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Shelukhin's Hofer distance and a symplectic cohomology barcode for contactomorphisms

Dylan Cant

Abstract

This paper constructs a persistence module of Floer cohomology groups associated to a contactomorphism of the ideal boundary of a Liouville manifold. The barcode (or, bottleneck) distance between the persistence modules is bounded from above by Shelukhin's Hofer distance. Moreover, the barcode is supported (i.e., has spectrum) on the lengths of translated points of the contactomorphism. We use this structure to prove various existence results for translated points and to construct spectral invariants for contactomorphisms which are monotone with respect to positive paths and continuous with respect to Shelukhin's Hofer distance. While this paper was nearing completion, the author was made aware of similar upcoming work by Djordjević, Uljarević, Zhang.

Shelukhin's Hofer distance and a symplectic cohomology barcode for contactomorphisms

Abstract

This paper constructs a persistence module of Floer cohomology groups associated to a contactomorphism of the ideal boundary of a Liouville manifold. The barcode (or, bottleneck) distance between the persistence modules is bounded from above by Shelukhin's Hofer distance. Moreover, the barcode is supported (i.e., has spectrum) on the lengths of translated points of the contactomorphism. We use this structure to prove various existence results for translated points and to construct spectral invariants for contactomorphisms which are monotone with respect to positive paths and continuous with respect to Shelukhin's Hofer distance. While this paper was nearing completion, the author was made aware of similar upcoming work by Djordjević, Uljarević, Zhang.
Paper Structure (51 sections, 12 theorems, 50 equations, 4 figures)

This paper contains 51 sections, 12 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of terms and statement of results
  3. Group of contactomorphisms
  4. Discriminant points, Reeb flow, and translated points
  5. Persistence module associated to a contactomorphism
  6. Bounding the barcode distance in terms of Shelukhin's Hofer distance
  7. Endpoints of bars lie in the spectrum
  8. Consequences of the main results
  9. A vanishing result for full symplectic cohomology
  10. Estimating the size of the spectrum
  11. Existence of two translated points when symplectic cohomology vanishes
  12. Existence of infinitely many translated points on ellipsoids
  13. Spectral invariants
  14. Orderability of ideal boundaries
  15. Non-orderable ideal boundaries and the boundary depth
  16. ...and 36 more sections

Key Result

Theorem 1.1

The barcode distance between $\mathfrak{B}_{\alpha}(\varphi_{1})$ and $\mathfrak{B}_{\alpha}(\varphi_{0})$ is bounded from above by Shelukhin's Hofer distance $\mathrm{dist}_{\alpha}(\varphi_{0},\varphi_{1})$, where the distance is measured in the universal cover.

Figures (4)

  • Figure 1: Defining the functor from $\Delta(\varphi_t)$ using the Serre fibration property; given $\varphi_{s,t}$, and lifts to the solid lines (i.e., $t=0$ and $s=0,1$) there is guaranteed to be some lift $\psi_{s,t}$. Evaluating at $t=1$ gives canonical homotopy class of paths from $\psi_{0,1}$ to $\psi_{0,1}$ (depending on $\varphi_{s,t}$).
  • Figure 2: Maximum principle for continuation cylinders. The equation is translation invariant outside the interval $[s_{0},s_{1}]$.
  • Figure 3: Cut-off function $\beta(s)$.
  • Figure 4: Choosing $s_0$ if $B$ is smaller than the length of the longest bar.

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Lemma 2.1
  • ...and 2 more