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Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules

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Abstract

We find robust obstructions to representing a Hamiltonian diffeomorphism as a full k-th power, \(k \ge 2\), and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer’s metric. Our approach is based on the theory of persistence modules applied in the context of filtered Floer homology. We present applications to geometry and dynamics of Hamiltonian diffeomorphisms.

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Notes

  1. This problem has been formulated by Misha Kapovich and L.P. at an Oberwolfach meeting in 2006.

  2. We remark that elsewhere in the literature (cf. [25, 26]) such fixed points are called simple, but this terminology would cause unnecessary confusion in the context of this paper.

  3. The original unsuccessful attempt of the authors was to use the BV operator.

  4. It was communicated to us by Paul Seidel.

  5. In order to prove that the map (4) is Lipschitz, one uses a deep isometry theorem between the interleaving distance on persistence modules and the bottleneck distance on barcodes, see [6].

  6. Our understanding of this picture appeared in discussions with Michael Usher and Jun Zhang. For its extension to general symplectic manifolds, we refer the reader to the paper [54] by Usher and Zhang. See also the paper [22] by Fraser.

  7. The term “two-dimensional” would cause confusion in our setting.

  8. This definition is a specific representative of the isomorphism class of limits of the indiscrete groupoid, namely a category with exactly one morphism between any two objects, formed by \(\{ HM ^{(a,b)}(f')\}_{f' \in \mathcal {F}(f)}\) and the continuation maps, rendering each two of these vector spaces canonically isomorphic. Note that this representative of the limit of this diagram is canonically isomorphic by a unique isomorphism to a similar representative of the limit of any of its full subdiagrams, namely subdiagrams with the same morphism sets as in the diagram between any two of their objects (since all the continuation maps are isomorphisms). This observation is useful in showing that this definition satisfies the properties of a r2p persistence module.

  9. This definition is a canonical representative of the isomorphism class in \(\mathcal {D}^S\) of limits in \(\mathcal {D}^S\) of the \(\mathcal {D}^S\)-valued diagram defined by Proposition 2.5.

  10. Here and below, we deal with certain transformations of loop spaces which naturally act on action functionals and on the Riemannian metrics on \(\mathcal {L}_\alpha M\) coming from loops of almost complex structures on M, thus inducing morphisms in Floer homology which are useful for our purposes. We call them diffeomorphisms since this way of thinking provides a right intuition for manipulating these Floer homological constructions. Incidentally, these transformations are genuine diffeomorphisms if understood in the sense of diffeology [31].

  11. We say that a perturbation \(H'\) of H is \(C^2\)-small if \(H'-H\) is a \(C^2\)-small function.

  12. In other words, these vector spaces and isomorphism maps form an indiscrete groupoid in \(\mathrm{Vect}_\mathcal {K}\).

  13. That is—we are considering a specific representative of the limit of the corresponding indiscrete groupoid.

  14. It would be interesting to compare this result with the work of Barannikov [5, 35, 36].

  15. That is the polynomial \(x^p-1 \in \mathcal {K}[x],\) which is separable by the assumption \(\mathrm{char}(\mathcal {K}) \ne p,\) splits over \(\mathcal {K}.\)

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Acknowledgments

We are grateful to Paul Seidel for his generous help with this paper (see Sect. 1.4 above). We thank Mohammed Abouzaid, Paul Biran, Strom Borman, Frédéric Bourgeois, Octav Cornea, Maia Fraser, Anatole Katok, Michael Khanevsky, Michael Usher, Claude Viterbo, Shmuel Weinberger and Jun Zhang for useful discussions. Our understanding of persistent homology profited from lectures by Yaniv Ganor, Asaf Kislev and Daniel Rosen at a guided reading course delivered by L.P. at Tel Aviv University. We thank all of them. Parts of this paper have been written during L.P.’s stay at the University of Chicago and ETH-Zürich, and E.S.’s stay at the Hebrew University of Jerusalem and CRM, University of Montreal. We thank these institutions for their warm hospitality. Preliminary results of this paper have been presented at symplectic workshops in the Lorentz Center, Leiden (Summer, 2014) and in the Clay Institute, Oxford (Fall, 2014). We are indebted to the organizers, Hansjörg Geiges, Viktor Ginzburg, Federica Pasquotto and Dominic Joyce, Alexander Ritter and Ivan Smith, respectively, for this opportunity. Finally, we thank the referee for a superb job including detecting and correcting several mistakes and suggesting a number of improvements.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Leonid Polterovich

  2. CRM, University of Montreal, Montreal, Canada

    Egor Shelukhin

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  1. Leonid Polterovich
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Correspondence to Leonid Polterovich.

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Leonid Polterovich: Partially supported by the European Research Council Advanced grant 338809.

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Polterovich, L., Shelukhin, E. Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Sel. Math. New Ser. 22, 227–296 (2016). https://doi.org/10.1007/s00029-015-0201-2

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