The Conley Conjecture

By V. Ginzburg. 2006

146 citations recorded by the source corpus.

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This theorem generalizes, for instance, a recent result of Hingston establishing the Conley conjecture for tori.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eb40f7949956ba6d594ab7e3bf79400a6e8f1875

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In Geometry and Topology

1. Basak Z. Gurel
2. V. Ginzburg
3. E. Shelukhin
4. P. Seidel
5. L. Polterovich
6. Michael Usher
7. D. Auroux
8. A. Oancea

In Statistical and Nonlinear Physics

1. Basak Z. Gurel
2. P. Seidel
3. E. Shelukhin
4. L. Polterovich
5. V. Ginzburg

Citations in the corpus

• Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem (2007, methodology): “[BPS], [Poz], [Gi9], [Vi]): Let F s be a family of Hamiltonians continuously parametrized by s 2 Œ0; 1 and such that † is an isolated set of periodic orbits for all F s .”
• Local Floer Homology and the Action Gap (2007, background): “mplexes of Hand G#Hshifting the action filtration by c and preserving the grading. This isomorphism sends a one-periodic orbit γof H to the one-periodic orbit ΦG(γ)(t) := ϕt G(γ(t)) of G#H; see, e.g., [Gi2] for more details. Hence, we obtain an isomorphism of Floer homology: HF(a,b) ∗ (H) ∼= −→ HF(a+”
• Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, methodology): “oops. The next two geometrical results discussed in this section concern the existence and extension of loops with specific properties and are used in the proof of Theorem 1.17. Here, we mainly follow [Gi3], taking into account cappings of periodic orbits. The first of these results asserts that every”
• Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “), all 1-periodic orbits in the conical part had the same action. The perturbation will destroy that, but only by a small amount. Hence, H (Q) is still a local Floer cohomology group, in the sense of [36, 22]. As in Lemma 3.4, one can replace HF(M;^ +) by HF (M; +) in (3.30), which yields (3.27). Th”
• Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules (2014, background): “[23, 22, 42]), which implies in particular that given φ there is an infinite subset Pφ ⊂ Z>0 of prime numbers such that for all p ∈ Pφ the diffeomorphism φ p has a contractible fixed point x which is not a fixed point of φ, or in other words φ has a contractible primitive p-periodic orbit x.”