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The action spectrum and \(C^0\) symplectic topology

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Our first main result states that the spectral norm \(\gamma \) on \( \mathrm {Ham}(M, \omega ) \), introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the \(C^0\) topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev-Shelukhin, we obtain the \(C^0\) continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism on any closed symplectic manifold of dimension greater than 2 having only one fixed point shows that the conjecture does not admit a direct generalization to the \( C^0 \) setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

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  1. We point out that the definition of Hamiltonian homeomorphisms that we adopt in this paper agrees with the one of Le Calvez [30], but differs from that of Müller and Oh [39] which is more restrictive.

  2. To be more precise, it is proven in [48] that the spectral norm is \(C^0\) continuous on surfaces.

  3. To be more precise, a barcode is a multi-set of intervals, i.e. a finite set of pairs (Im), where I is an interval and m is a positive integer. However, to keep the exposition simple, we use the notation \(\{I_j\}_{1 \leqslant j \leqslant N}\), keeping in mind that the list of \(I_j\)’s may have repetitions.

  4. Here, \(\cap \) refers to the intersection product in homology. Cup length can be equivalently defined in terms of the cup product in cohomology.

  5. The original version of the Arnold conjecture, in which the lower bound for the number of fixed points is predicted to be the minimal number of critial points of a smooth function on M, has also been established on ashperical manifolds; see [44].

  6. The set of spectral invariants of this Hamiltonian homeomorphism coincides with the spectral invariants of a \(C^2\)–small Morse function. Hence, their count is at least the cup length of the manifold. This is perhaps an indication that, on symplectic manifolds of dimension at least four, one cannot define the notion of action for fixed points of an arbitrary Hamiltonian homeomorphism. See Remark 20 in [2].

  7. The original question was posed in the two-dimensional setting, whence the use of the word “disk”.

  8. The almost conjugacy relation may be characterized by the following universal property: \(\varphi \sim \psi \) if and only if \(f(\varphi ) = f(\psi )\) for any continuous function \(f: {\overline{\mathrm {Ham}}}(M, \omega ) \rightarrow Y\) such that f is invariant under conjugation and Y is a Hausdorff topological space.

  9. A notion very closely related to that of almost conjugacy, called \(\chi \)–equivalence, arises naturally in the study of surface group actions on the circle; see [34] and references therein.

  10. See [36] for the construction of these invariants on general symplectic manifolds.

  11. Here, the subscript “LS” refers to “Lusternik-Schnirelman” since this quantity is related to the so-called Lusternik-Schnirelman theory; this is discussed in further details in Sect. 5.1.

  12. To find such K, one can for instance take a sufficiently fine triangulation of M, and let K be the union of all top-dimentional simplices that have non-empty intersection with \(f^{-1}(\max (f))\).

  13. By using the well-known fact [45] that on a closed symplectically aspherical manifold, the action spectrum of a contractible normalized Hamiltonian loop is \( \{ 0 \} \).

  14. By residual subset we mean a countable intersection of dense open subsets.


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The generalization of the Arnold conjecture presented in this paper came to life after our realization that the methods used by Wyatt Howard in [21] could be adapted to non-smooth settings. We thank him for helpful communications. We would like to thank Pierre-Antoine Guihéneuf, Helmut Hofer, Rémi Leclercq, Frédéric Le Roux, Nicolas Vichery and Claude Viterbo for helpful conversations. We are also grateful to the anonymous referee for his very careful reading and his valuable comments. Lastly, we thank Egor Shelukhin for bringing the following to our attention: (a variant of) Inequality (1) and the fact that it implies \(C^0\) continuity of barcodes; these results appear in [26]. SS This paper was partially written during my stay at the Institute for Advanced Study in the academic year 2016-2017. I greatly benefited from the lively research atmosphere of the IAS and would like to thank the members of the School of Mathematics for their warm hospitality. VH is partially supported by the ANR project “Microlocal” ANR-15-CE40-0007. LB is partially supported by ISF grants 1380/13 and 2026/17, by the ERC Starting grant 757585, and by the Alon fellowship.

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Correspondence to Vincent Humilière.

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Fixed and periodic points of \(C^0\) generic Hamiltonian homeomorphisms

Fixed and periodic points of \(C^0\) generic Hamiltonian homeomorphisms

Estimating the number of fixed points of generic (in a \(C^1\) sense) Hamiltonian diffeomorphisms has been a central problem in symplectic topology over the past 30 years. The construction of Floer homology implies that this number is bounded below by the sum of the Betti numbers of the manifold. It is therefore natural to ask if similar estimates hold for \(C^0\) generic Hamiltonian homeomorphisms. It turns out that the situation is dramatically different, as the next proposition shows.

Proposition A.1

Let \((M, \omega )\) be any closed symplectic manifold. There exists a residualFootnote 14 subset \({\mathcal {U}}\) of \({\overline{\mathrm {Ham}}}(M,\omega )\) such that every element in \({\mathcal {U}}\) has infinitely many fixed points.

In the case of a symplectic surface \((\Sigma , \omega )\), a stronger result holds: Generically in \({\overline{\mathrm {Ham}}}(\Sigma ,\omega )\), the set of fixed points is a Cantor set; see [16]. The proof uses tools that are not available in higher dimensions. Therefore it is not clear to us if this result extends to higher dimensions or not.

The proof of Proposition A.1 will follow easily from the Lefschetz index theory. The Lefschetz index is an integer associated to an isolated fixed point of a continuous map. Of its properties, we will use the following (see e.g. [24], Chapter 2, Section 8.4):

  1. (a)

    The index of a non-degenerate fixed point of a diffeomorphism is either 1 or \(-1\).

  2. (b)

    Fixed points with non-zero index are \(C^0\) stable. More pecisely, if \(x_0\) is an isolated fixed point of a continuous map f, then there exists a \(C^0\) neighborhood of f such that every map in this neighborhood admits a fixed point near \(x_0\).

We are now ready for the proof.


Let \(\phi \) be a non-degenerate Hamiltonian diffeomorphism which admits at least N fixed points. According to Property (a) above, all its fixed points have non-zero index, hence by Property (b), there exists a \(C^0\) open subset \(U_\phi \subset {\overline{\mathrm {Ham}}}(M,\omega )\) such that every element in \(U_\phi \) has at least N fixed points.

Now, it is well known that non-degenerate Hamiltonian diffeomorphisms are dense in \(\mathrm {Ham}(M,\omega )\), hence in \({\overline{\mathrm {Ham}}}(M,\omega )\). Furthermore, given a non-degenerate Hamiltonian diffeomorphism \(\psi \), we can always perform a local modification near one of its fixed points (which are known to exist by Floer theory), to construct a new Hamiltonian diffeomorphism, \(C^0\) close to \(\psi \) and admitting at least N fixed points. To perform this construction near a fixed point x, choose N distinct points \(x_1,\dots , x_N\) in a small ball B around x such that \(\psi (x_i)\in B\) for all \(i\in B\), and pick a diffeomorphism \(\chi \) supported in B with \(\chi (\psi (x_i))=x_i\) for all \(i=1,\dots , N\). Then, \(\chi \circ \psi \) admits all \(x_1,\dots , x_N\) as fixed points. Moreover, the \(C^0\) distance between \(\chi \circ \psi \) and \(\psi \) can be made arbitrarily small by shrinking B. Finally, we can perturb \(\chi \circ \psi \) to make all fixed points \(x_1,\dots , x_N\) non-degenerate.

As a consequence, the set \(V_N\) of non-degenerate Hamiltonian diffeomorphisms admitting at least N fixed points is dense in \({\overline{\mathrm {Ham}}}(M,\omega )\).

It follows immediately that the set \({\mathcal {U}}_N=\bigcup _{\phi \in V_N}U_\phi \) is a dense open subset and that all its elements have at least N fixed points. Finally, \({\mathcal {U}}=\bigcap _{N>0}U_N\) is a residual subset all of whose elements have infinitely many fixed points. \(\square \)

We end the paper with a brief discussion of the Conley conjecture on closed and symplectically aspherical manifolds. The conjecture, which was proven in [13, 18], states that a Hamiltonian diffeomorphism must have infinitely many periodic points. When it comes to Hamiltonian homeomorphisms the conjecture was proven on surfaces in [30]. In dimensions four and higher, we are neither able to prove it nor are we able to adapt our counterexample from [2] to disprove it. However, we believe that one should be able to adapt the proof of Proposition A.1 to show that, for a \(C^0\) generic Hamiltonian homeomorphism \(\phi \), the set \({\mathcal {P}}_k(\phi )\) consisting of periodic points of period k, must be infinite for infinitely many values of k.

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Buhovsky, L., Humilière, V. & Seyfaddini, S. The action spectrum and \(C^0\) symplectic topology. Math. Ann. 380, 293–316 (2021).

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