Symplectic Geometry: A Festschrift in Honour of Claude Viterbo’s 60th Birthday

Contents
Dedication
Symplectically convex and symplectically star-shaped curves: a variational problem
Abstract
1. Introduction
2. Examples of symplectically convex and symplectically star-shaped curves
3. Toward the solution of the variational problem
4. Infinitesimal rigidity of multiple conics as critical curves
5. Second-order deformations of conics
6. The case of
7. Pictures and open problems
Acknowledgements
Appendix
References
C0-robustness of topological entropy for geodesic flows
Abstract
1. Introduction
1.1. Context
1.2. Results, strategy, and layout of the paper
1.3. Related developments
1.4. Setup and definitions
Entropies
Length, energy, area and loop spaces
Robustness
A preliminary robustness lemma
2. A robust version of the Denvir–MacKay theorem
2.1. Proof of item (1) of Theorem 5 in case γ0 is non-degenerate
2.2. Proof of item (1) of Theorem 5 in case γ0 is degenerate
2.3. A lower bound on topological entropy in terms of surface area
3. Robustness of volume entropy and hyperbolic geometry
4. Robustness from intersection patterns of a family of non-contractible geodesics on the two-torus
4.1. A definition of separation for lifts of two freely homotopic loops
4.2. The two-torus and an intersection pattern
4.3. Robustness of entropy via ribbons
4.4. Ribbons exist for Cinfty generic metrics
5. Robustness by retractable neck on general manifolds
Appendix A. Robustness of non-degenerate length spectrum
References
Bifurcations of balanced configurations for the Newtonian n-body problem in R4
Abstract
1. Introduction
2. S-balanced configurations in the n-body problem
3. A brief recap on the spectral flow in finite dimension
4. Bifurcations of collinear S-balanced configurations
5. Some explicit examples for n=3
5.1. The continuation method
5.2. Results of the computations
5.3. Final comments
Acknowledgements
References
Relative Hofer–Zehnder capacity and positive symplectic homology
Abstract
1. Periodic orbits for Hamiltonian systems
1.1. From calculus of variation to pseudoholomorphic curves
1.2. Capacities and symplectic homology for Liouville domains
2. Statement of main results
Organization of the paper
3. Floer homologies and symplectic capacities
3.1. Hamiltonian Floer homology
3.2. Symplectic homology
3.3. Capacities from Floer and symplectic homology
Proof of Theorem 1.3
4. Proofs of the main results
4.1. Proof of Theorem 2.2
4.2. Proof of Theorem 2.9(a)
4.3. Proof of Theorem 2.9(b)
4.4. Proof of Corollary 2.8
Acknowledgements
Appendix: The monotonicity of the systole of convex Riemannian two-spheres (by Alberto Abbondandolo1 and Marco Mazzucchelli)2
References
An Arnold-type principle for non-smooth objects
Abstract
1. Introduction and main results
Organization of the paper
2. Preliminaries from symplectic geometry
2.1. Symplectic and Hamiltonian homeomorphisms
2.2. Hofer's distance
3. Preliminaries on spectral invariants
3.1. Min–max critical values and Lusternik–Schnirelmann theory
3.2. Spectral invariants for Lagrangians
3.3. Spectral invariants for Legendrians via generating functions
4. C0 Lagrangians, proof of Theorem 1.1 and Proposition 1.2
4.1. Spectral invariants for C0 Lagrangians
4.2. Proof of Theorem 1.1
4.3. Proof of Proposition 1.2
5. Hausdorff limits of Legendrians and proof of Theorem 1.5
Acknowledgements
References
Quantitative h-principle in symplectic geometry
Abstract
1. Introduction
2. Quantitative h-principle for isotropic discs
2.1. Standard h-principle for subcritical isotropic embeddings
2.2. Proof of theorem 1
3. Action of symplectic homeomorphisms on symplectic submanifolds
3.1. Taking a symplectic disc to an isotropic one
3.2. Relative Eliashberg–Gromov mathbbmathcalC0-rigidity
Acknowledgements
References
On symplectomorphisms and Hamiltonian flows
Abstract
1. Introduction
2. Preliminary background. Time one flows of time dependent Hamiltonians.
3. Symplectomorphisms as Lagrangian submanifolds
4. Iterations and Euler midpoint estimates in action
4.1. Setting and definitions
4.2. Main estimate and its iterations
Acknowledgements
A Synopsis on the implicit Euler midpoint scheme
A.1 Basic theory
A.2 Estimates
References
Lagrangian skeleta and plane curve singularities
Abstract
1. Introduction
1.1. Main results
2. Lagrangian skeleta for isolated singularities
2.1. The legendrian link of an isolated singularity
2.2. A'Campo's divides and their conormal lifts
2.3. Proof of Theorem 1.1
2.4. Lagrangian skeleta
3. Augmentation stack and the cluster algebra of Fomin–Pylyavskyy–Shustin–Thurston
4. A few computations and remarks
4.1. Cal-skeleta for An-singularities
5. Structural conjectures on Lagrangian fillings
5.1. Some questions
Acknowledgements
References
Reeb chords of Lagrangian slices
Abstract
1. Introduction
2. What does a Lagrangian slice look like?
3. Deformation of Liouville vector fields near the boundary
4. Collarability of Lagrangian slices
5. Reeb chords of fillable slices
6. Concluding remarks
Acknowledgements
References
Basic facts and naive questions
Abstract
1. Subharmonic bifurcations in real or complex dimension one
1.1. The period doubling bifurcation
1.2. Subharmonic bifurcations, holomorphic case
1.3. Opening Pandora's box
2. Subharmonic bifurcations, Arnold tongues and KAM circles
2.1. Subharmonic bifurcations in real dimension two
2.2. Arnold tongues
2.3. Opening Pandora's box wider
3. Higher dimensions
3.1. Statement of the hypotheses
3.2. Periodic orbits
3.3. Passing to the limit in the holomorphic case?
3.4. Passing to the limit in the smooth case?
4. Comments and references
References
Another look at the Hofer–Zehnder conjecture
Abstract
1. Introduction and main results
1.1. Introduction
1.2. Shelukhin's theorem
2. Preliminaries
2.1. Conventions and notation
2.2. Equivariant Floer cohomology and the pair-of-pants product
3. Floer graphs
3.1. Main result
3.2. Implications and the proof of Theorem 1.1
4. A few words about the shortest bar
5. Proof of Theorem 3.1 and further remarks
5.1. Proof of Theorem 3.1
5.2. Degenerate case
Acknowledgements
References
Higher symplectic capacities and the stabilized embedding problem for integral elllipsoids
Abstract
1. Introduction
1.1. The main results
1.2. Applications and remarks
1.2.1. Steps and the rescaled embedding function.
1.2.2. The first step
1.2.3. The case b = 1
1.2.4. The other parity
Structure of the note
2. New capacities
2.1. The first approximation
2.2. Behavior under stabilization
2.3. The naive chain complex
2.4. Input from symplectic field theory
2.5. From spectral invariants to capacities
2.6. The case of ellipsoids
3. Optimal embeddings
3.1. The main theorems
3.1.1. The construction
3.1.2. Some obstructions
3.1.3. The proofs
3.2. The rescaled embedding function
3.3. The first step
3.4. The other parity
4. Discussion
4.1. Beyond the rescaled function
4.2. The opposite parity
4.3. The region from b = 1 to b = 2
4.4. A combinatorial rule?
Acknowledgements
References
Closed geodesics on reversible Finsler 2-spheres
Abstract
1. Introduction
1.1. The curve shortening semi-flow
1.2. Closed geodesics on Finsler 2-spheres
1.3. Organization of the paper
2. The curve shortening semi-flow
2.1. The evolution equation
2.2. The anti-gradient of the length
2.3. Short-time existence
2.4. Long-time existence
2.5. Linfty bounds on Vγ
2.6. Compactness
3. Existence of simple closed geodesics
3.1. Lusternik–Schnirelmann theory
3.2. Topology of the space of embedded circles on the 2-sphere
4. Critical point theory of the energy functional
4.1. The energy functional
4.2. The Morse index and nullity
4.3. Local homology
5. Infinitely many closed geodesics
5.1. The Birkhoff map
5.2. Periodic points of twist maps
5.3. Hingston's theorems
5.4. Bangert's theorem
Acknowledgements
References
Families of Legendrians and Lagrangians with unbounded spectral norm
Abstract
1. Introduction and results
1.1. Why the proofs of uniform bounds fail for Legendrians
2. Background
2.1. Contact geometry of jet-spaces and contactisations
2.1.1. The cotangent bundle and jet-space
2.1.2. The punctured torus
2.2. Barcode of a filtered complex and notions from spectral invariants
2.3. Outline of Floer homology and generating family homology for Legendrians
2.4. Floer complex as the linearised Chekanov–Eliashberg algebra
2.5. Maslov potential and grading
3. Examples that exhibit unbounded spectral norms
3.1. Legendrian isotopy of the unknot (Proof of Part (2) of Theorem A)
3.2. Legendrian isotopy of the zero-section (Proof of Part (1) of Theorem A)
3.3. Hamiltonian isotopy on the punctured torus (Proof of Theorem D)
4. Proof of Theorem C
Acknowledgements
References
Legendrian persistence modules and dynamics
Abstract
1. Introduction and main results
1.1. Interlinking
1.2. The pool of contact manifolds
1.3. Sample applications: chords of symplectic Hamiltonians
1.4. Sample applications: contact dynamics
1.4.1. Contact interlinking
1.4.2. Beyond interlinking
1.4.3. Chords of contact Hamiltonians
1.4.4. Contact flows with large conformal factor
1.5. Scheme of the proof: homologically bonded pairs
1.6. Plan of the paper
2. A modified pb+-invariant and Hamiltonian chords
2.1. Chords of smooth vector fields
2.2. Poisson bracket invariant
3. Persistence modules
4. Legendrian contact homology
4.1. Exact Lagrangian cobordisms
4.2. Chekanov–Eliashberg algebra and the corresponding persistence modules
4.3. Morphisms of persistence modules defined by Lagrangian cobordisms
4.4. An invariant of two-part Legendrians via LCH persistence modules
4.5. A lower bound on pb+ via LCH persistence modules
5. Applications to contact dynamics
5.1. Largeness of the conformal factor of t
5.2. Existence of chords of h from Λ0 to Λ1
6. The case of J1 Q
7. The case of ST* mathbbRn
Acknowledgements
References
A Lagrangian Klein bottle you can't squeeze
Abstract
1. Introduction
2. The minimal genus question
2.1. Review
2.2. S2timesS2
3. Nonsqueezing
3.1. Statement
3.2. Mohnke's almost complex structure
3.3. Neck-stretching
3.4. SFT limit analysis
Acknowledgements
References
Construction of a linear K-system in Hamiltonian Floer theory
Abstract
1. Introduction
2. Floer's equation and moduli space of solutions
3. Stable map compactification of Floer's moduli space
3.1. Definition of mathcalMell(X,H;α-,α+)
3.2. Topology on mathcalMell(X,H;α-,α+)
4. Construction of Kuranishi structure
4.1. Statement
4.2. Obstruction bundle data
4.3. Smoothing singularities and ε-closeness
4.4. Kuranishi chart
4.5. Coordinate change
5. Compatibility of Kuranishi structures
5.1. Outer collar
5.2. Proof of Proposition 5.5 I: Obstruction space with outer collar
5.3. Proof of Proposition 5.5 II: Kuranishi chart and coordinate change
6. Construction of morphism
6.1. Statement
6.2. Proof of Theorem 6.4 (1)(2): Kuranishi structure
6.3. Proof of Theorem 6.4 (3)(4): Kuranishi structure with outer collar
7. Construction of homotopy
8. Composition of morphisms
8.1. Statement
8.2. Proof of Theorem 8.6 (1): Kuranishi structure
8.3. Proof of Theorem 8.6 (2): Kuranishi structure with outer collar
9. Well-definedness of Hamiltonian Floer cohomology
10. Calculation of Hamiltonian Floer cohomology
References
What does a vector field know about volume?
Abstract
1. Introduction
2. Dimension three
3. Geodesible vector fields and taut foliations
4. Basic cohomology
5. The Euler class of a geodesible vector field
6. Seifert fibred 3-manifolds
7. The theorems of Gauß–Bonnet and Poincaré–Hopf
8. Transversely holomorphic foliations and the Bott invariant
9. Global surfaces of section
10. Contact forms with the same Reeb vector field
11. Orbit equivalence
Acknowledgements
References
On the symplectic fillings of standard real projective spaces
Abstract
1. Introduction
2. The moduli space of lines
2.1. The smooth stratum
2.2. The compactified moduli space
3. Proof of the main theorem
3.1. Degree of the evaluation map
3.2. Decomposition of the line
4. Fundamental group of semipositive fillings
5. Yet another proof of the Eliashberg-Floer-McDuff theorem
References
On curves with the Poritsky property
Abstract
1. Introduction and main results
1.1. The Poritsky property for string construction
1.2. The Poritsky property for outer billiards and area construction
1.3. Coincidence of the Poritsky and Lazutkin lengths
1.4. Unique determination by 4-jet
2. Background material from Riemannian geometry
2.1. Normal coordinates and equivalent definitions of geodesic curvature
2.2. Angular derivative of exponential mapping and the derivatives dAdC, dBdC
2.3. Geodesics passing through the same base point: azimuths of tangent vectors at equidistant points
2.4. Geodesic-curvilinear triangles in normal coordinates
3. The string foliation: proof of Theorem 1.3
3.1. Finite smoothness lemmas
3.2. Proof of Theorem 1.3
4. Billiards on surfaces of constant curvature: proofs of Proposition 1.6 and Theorem 1.7
4.1. Proof of Proposition 1.6
4.2. Preparatory coboundary property of length ratio
4.3. Conics and Ceva's Theorem on surfaces of constant curvature: proof of Theorem 1.7
5. Case of outer billiards: proof of Theorem 1.12
6. The function L(A,B) and the Poritsky–Lazutkin parameter: proofs of Theorems 1.16 and 1.15, and Corollaries 1.17 and 1.18
7. Symplectic generalization of Theorem 1.15
7.1. Symplectic properties of billiard ball map
7.2. Families of billiard-like maps with invariant curves: a symplectic version of Theorem 1.15
7.3. Modified Lazutkin coordinates and asymptotics
7.4. Proof of Theorem 7.10
7.5. Deduction of Theorem 1.15 (case C6) from Theorem 7.10
8. Osculating curves with the string Poritsky property: proof of Theorem 1.19
8.1. Cartan distribution, a generalized version of Theorem 1.19 and plan of the section
8.2. Differential equations in jet spaces and the Main Lemma
8.3. Comparison of functions L(0,t) and Λ(t) for osculating curves
8.4. Dependence of functions L(0,t) and Λ(t) on the metric
8.5. Taylor coefficients of Λ(t): analytic dependence on jets
8.6. Proof of Lemma 8.6
8.7. Proof of Theorems 8.4 and 1.19
Acknowledgements
References
Correction to: On curves with the Poritsky property
Correction to: J. Fixed Point Theory Appl. (2022) 24:35 https://doi.org/10.1007/s11784-022-00948-7
Examples around the strong Viterbo conjecture
Abstract
1. Introduction
Organization of the paper
2. Toric domains
3. The first equivariant capacity
4. ECH capacities
5. A family of non-monotone toric examples
6. The first Ekeland–Hofer capacity
Acknowledgements
References
Global surfaces of section with positive genus for dynamically convex Reeb flows
Abstract
1. Introduction and main results
2. Preliminaries
2.1. Periodic orbits, asymptotic operators and Conley–Zehnder indices
2.2. Pseudoholomorphic curves in symplectizations
2.3. Asymptotic cycles
3. Proof of Theorem 1.1
4. Proof of Theorem 1.4
Acknowledgements
References
ECH capacities and the Ruelle invariant
Abstract
1. Introduction
1.1. Asymptotics of ECH capacities
1.2. The Ruelle invariant
1.3. Results for toric domains
1.4. Outline of the rest of the paper
2. The Ruelle invariant of toric domains
3. Bounds on ECH capacities of toric domains
3.1. The Ruelle invariant and the weight expansion
3.2. An estimate from the weight expansion
3.3. Lattice point estimates
3.4. Completing the proof of the main theorem
4. Improving the exponent in the general case
5. Heuristics for the conjecture
5.1. Facts
5.2. A new definition
5.3. Heuristics
Acknowledgements
References
Capacities of billiard tables and S1-equivariant loop space homology
Abstract
1. Introduction
2. Definition of capacities
3. Properties of capacities
3.1. Conformality and monotonicity
3.2. Nontriviality
3.3. Periodic billiard trajectory
3.4. Growth of cS1k
4. Relation to symplectic capacities
5. Capacities of rectangles
5.1. Reduction to key lemmas
5.2. Bott–Morse theory on free loop spaces
5.2.1. Setting
5.2.2. Variational method for Lagrangian action functional
5.3. Proofs of Lemmas 5.3 and 5.4
5.4. Application to periodic billiard trajectories
6. Questions
Acknowledgements
References
Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities
Abstract
1. Introduction
2. Examples
2.1. In dimension two
2.2. Ellipsoids
2.3. Smooth starshaped toric domains
2.4. Starshaped domains with the symmetric ratio slightly bigger than one
2.5. Bordeaux-bottle-shaped hypersurfaces of arbitrarily large symmetric ratio
2.6. Hypersurfaces of arbitrarily large symmetric ratio in Hamiltonian systems
3. Closed-open maps
3.1. Symplectic homology
3.1.1. Admissible Hamiltonians
3.1.2. Chain complex
3.1.3. Symplectic homology
3.1.4. Action filtration
3.1.5. Tautological exact sequences
3.2. Wrapped Floer homology
3.2.1. Chain complex
3.2.2. Filtered wrapped Floer homology
3.3. Closed-open maps
3.3.1. Floer data
3.3.2. Floer chimneys
3.3.3. Filtered closed-open maps
3.3.4. Without absolute grading
4. Floer homology capacities
4.1. SH capacity
4.2. HW capacity
4.3. Proof of Theorem 1.3
5. Real symplectic capacities
Acknowledgements
References
Diffeomorphism type via aperiodicity in Reeb dynamics
Abstract
1. Introduction
2. Aperiodicity and boundary shape
2.1. A model
2.2. Fibrewise shaped
2.3. Standard near the boundary
2.4. Main theorem
2.5. Comments on Theorem 2.1
3. The degree method
3.1. Completion via gluing
3.2. Filling by holomorphic discs
4. Standard holomorphic discs
4.1. The contactisation
4.2. Liouville manifold and potential
4.3. The symplectisation
4.4. The Niederkrüger transform
4.5. Class independence
5. Symplectic potentials on cotangent bundles
5.1. Dual connection
5.2. Orthogonal splitting
5.3. Taming structure
5.4. Potentials
5.5. Interpolating geodesic and normalised geodesic flow
6. A boundary value problem
6.1. An almost complex structure
6.2. The moduli space
6.3. Uniform energy bounds
6.4. Maximum principle
6.5. Integrated maximum principle
7. Compactness
8. Transversality
8.1. Maslov index
8.2. Simplicity
8.3. Variable boundary conditions
8.4. Admissible functions
8.5. Linearised Cauchy–Riemann operator
8.6. Lifting topology
9. The homotopy type
9.1. Homology type and fundamental group
9.2. A cobordism
9.3. Infinite coverings
Acknowledgements
References
Conservative surface homeomorphisms with finitely many periodic points
Abstract
1. Introduction
1.1. The case of the sphere
1.2. The case of the torus
1.3. The case of high genus
1.4. Plan of the article
2. Definitions, notations and preliminaries
2.1. Loops and paths
2.2. Homeomorphisms of hyperbolic surfaces
2.3. Non wandering homeomorphisms
2.4. Poincaré–Birkhoff Theorem
2.5. A fixed point theorem for a planar homeomorphism
2.6. Some forcing results
3. Dehn twist maps
4. Homeomorphisms isotopic to the identity
5. The case of the torus
References
The Anosov–Katok method and pseudo- rotations in symplectic dynamics
Abstract
1. Introduction
Organization of the paper
2. The conjugation method of Anosov and Katok
2.1. General scheme
2.2. Why is f a pseudo-rotation?
2.3. Transitivity
2.4. Pseudo-rotations with minimal number of ergodic measures
3. Preliminaries
3.1. Preliminaries on symplectic geometry
3.2. Preliminary lemmas
(a) The transversality lemma
(b) The thickening lemma
(c) The stability lemma
(d) The transportation lemmas
3.3. Proofs of the lemmas
(a) Proof of lemma 5: Transversality lemma
(b) Proof of lemma 6: Thickening lemma
(c) Proof of lemma 7: Stability lemma
(d) Proof of lemma 8: Transportation lemma
4. Proof of proposition 4
4.1. Equidistribution boxes
4.2. Transportation boxes
4.3. Small boxes
4.4. From boxes to proposition 4
Acknowledgements
References
Contact geometry in the restricted three-body problem: a survey
Abstract
1. Introduction
2. Basic concepts
2.1. Symplectic geometry
2.2. Contact geometry
2.3. Open book decompositions
2.4. Global hypersurfaces of section
2.5. Examples of adapted dynamics
2.6. Geodesic flow on Sn and the geodesic open book
2.7. Double cover of SS2
2.8. Magnetic flows and quaternionic symmetry
2.9. The magnetic open book decompositions
3. The three-body problem
4. Moser regularization
4.1. Stark–Zeeman systems
4.2. Levi–Civita regularization
4.3. Kepler problem
5. Historical remarks
6. Contact geometry in the restricted three-body problem
6.1. Non-perturbative methods: holomorphic curves
7. Holomorphic curve techniques on the spatial CR3BP
7.1. Step (1): Global hypersurfaces of section
7.2. Step (2): Fixed-point theory of Hamiltonian twist maps
7.3. Alternative approach: dynamics on moduli spaces
8. Conclusion and further discussion
Acknowledgements
References
A generalized Poincaré–Birkhoff theorem
Abstract
1. Introduction
1.1. Poincaré–Birkhoff theorem, and the planar restricted three-body problem
Fixed-point theory of Hamiltonian twist maps
1.2. The Hamiltonian twist condition
1.3. Index growth
1.4. Fixed-point theorems
1.5. Sketch of the proof
1.6. Remarks on the twist condition and generalizations
2. Motivation and background
Hypersurfaces of section, return maps, and open books
3. Preliminaries on symplectic homology
3.1. Liouville domains and Hamiltonian dynamics
3.2. Conley–Zehnder index, Robbin–Salamon index, and mean index
3.3. Hamiltonian Floer homology and symplectic homology
3.4. Continuation maps and symplectic homology
3.5. Degenerate Hamiltonians and local Floer homology
3.6. Spectral sequence
3.7. Index-definiteness and grading
4. Proof of the Generalized Poincaré–Birkhoff Theorem
4.1. Filtration by homotopy class
4.2. Filtration by index
Acknowledgements
Appendix A: Hamiltonian twist maps: examples and non-examples
A.1. Examples
A.2. Non-examples: Katok examples
A.3. Relation with the Katok examples
Appendix B: Symplectic homology of surfaces
Appendix C: On symplectic return maps
Appendix D: Strong convexity implies strong index-positivity
Appendix E: Strongly index-definite symplectic paths
References
Covariant constancy of quantum Steenrod operations
Abstract
1. Introduction
2. A bit of equivariant (co)homology
3. Basic moduli spaces
4. Quantum steenrod operations
5. Proof of Theorem 1.4
6. Computations
Acknowledgements
References
An algebraic approach to the algebraic Weinstein conjecture
Abstract
Acknowledgements
References
Quantum cohomology as a deformation of symplectic cohomology
Abstract
1. Introduction
1.1. Geometric setup
1.2. Quantum cohomology is a deformation of symplectic cohomology
1.3. Rigidity results
1.4. Floer theory conventions
1.5. Outline of proofs
1.5.1. Theorem B
1.5.2. Theorem C
1.5.3. Theorem D
1.6. Conjectures
1.6.1. Filtration on QH*(M;Λ)
1.6.2. Analogue of Theorem C in the absence of Hypothesis A
1.6.3. Maurer–Cartan element
1.6.4. Mirror symmetry in the log Calabi–Yau case
1.7. Examples
1.7.1. Fano toric varieties
1.7.2. Skeleta in S2
1.7.3. The case M=S2, D= a point
1.7.4. The quadric in mathbbCmathbbP2
1.7.5. Fano hypersurfaces
1.8. Outline
2. Symplectic divisors
2.1. Basics
2.2. Systems of commuting Hamiltonians
2.3. Adapted Liouville one-forms
2.4. Admissibility
3. Quantum, Hamiltonian Floer, and symplectic cohomology
3.1. Quantum and Hamiltonian Floer cohomology
3.2. Relative symplectic cohomology
3.3. Towards the symplectic cohomology of the divisor complement
3.4. Positivity of intersection
4. Special Hamiltonian
4.1. Overview of the construction of ρR
4.2. Construction of UDImax, rImax, DImax
4.3. Construction of tildeYRI
4.4. Construction of tildeρRI
4.5. Construction of ρR
4.6. The Hamiltonian and its orbits
4.7. Perturbing to achieve nondegeneracy
4.8. Action computation
4.9. Index computation
5. Proofs
5.1. Properties of tildeρRI
5.2. Proof of Theorem B
5.3. Proof of Theorem C
5.4. Proof of Theorem D
Acknowledgements
A. Algebraic background
A.1. Filtration maps
A.2. Quasi-isomorphic subcomplexes of the telescope
A.3. Completed telescopes
A.4. Homotopy inverse limit
References
A symplectic embedding of the cube with minimal sections and a question by Schlenk
Abstract
1. The main result
2. Proofs of Theorem 4 and of Corollary 5
Acknowledgements
References