The Conley Conjecture and Beyond

Ginzburg, Viktor L.; Gürel, Başak Z.

doi:10.1007/s40598-015-0017-3

Cited by 38 publications

(35 citation statements)

References 126 publications

(278 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, the conjecture holds for all symplectic CY manifolds, [GG09,He], and all negative monotone symplectic manifolds, [CGG,GG12]. (See also [FrHa,Gi10,Hi,LeC,SZ] for some milestone intermediate results and [GG15] for a general survey and further references.) One can also show that N ≤ 2n when M admits a pseudo-rotation, where N is the minimal Chern number of M , although it is not yet known if the Conley conjecture holds whenever N > 2n.…”

mentioning

confidence: 97%

“…One can also define a pseudo-rotation as a Hamiltonian diffeomorphism with finitely many periodic points. These are the so-called perfect Hamiltonian diffeomorphisms studied in the context of the Conley conjecture; see, e.g., [CGG,GG15] and references therein. As follows from the results in [Fr92,Fr96,FrHa,LeC], in dimension two this definition is equivalent to the one adopted here.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Hamiltonian pseudo-rotations of projective spaces

Ginzburg

Gürel

2018

Invent. math.

Self Cite

View full text Add to dashboard Cite

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of CP n with the minimal possible number of periodic points (equal to n + 1 by Arnold's conjecture), called here Hamiltonian pseudorotations. We prove several results on the dynamics of pseudo-rotations going beyond periodic orbits, using Floer theoretical methods. One of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points, partially extending a theorem of Le Calvez and Yoccoz and of Franks and Misiurewicz to higher dimensions. The other is a strong variant of the Lagrangian Poincaré recurrence conjecture for pseudo-rotations. We also prove the C 0 -rigidity of pseudo-rotations with exponentially Liouville mean index vector. This is a higher-dimensional counterpart of a theorem of Bramham establishing such rigidity for pseudo-rotations of the disk.

show abstract

mentioning

confidence: 97%

mentioning

confidence: 99%

Hamiltonian pseudo-rotations of projective spaces

Ginzburg

Gürel

2018

Invent. math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Other applications of cylindrical contact homology are a contact version of the nonsqueezing theorem [67,85], and proofs of cases of the Weinstein conjecture and of versions of the Conley conjecture for Reeb flows, see the survey [94]. Rational SFT (namely SFT where all J-curves are punctured spheres) is used in [66] and [48] to find obstructions to the existence of Lagrangian submanifolds in symplectic manifolds that contain many holomorphic spheres (like complex projective space).…”

Section: Application 2: Hamiltonian and Contact Closing Lemmasmentioning

confidence: 99%

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

View full text Add to dashboard Cite

Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer's seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry. 14 4.

show abstract

“…The second application is along the lines of the Conley conjecture (see [GG15] or [Vi92,Prop. 4.13]) and concerns the number or the growth of simple periodic orbits of compactly supported Hamiltonian diffeomorphisms.…”

Section: Stable Displacementmentioning

confidence: 99%

On the filtered symplectic homology of prequantization bundles

Ginzburg

Shon

2018

Int. J. Math.

Self Cite

View full text Add to dashboard Cite

We study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit -the linking number filtration -and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e., the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.

show abstract

The Conley Conjecture and Beyond

Cited by 38 publications

References 126 publications

Hamiltonian pseudo-rotations of projective spaces

Hamiltonian pseudo-rotations of projective spaces

Floer Homologies, with Applications

On the filtered symplectic homology of prequantization bundles

Contact Info

Product

Resources

About