The action spectrum and $C^0$ symplectic topology

Abstract

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.

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 residual¹⁴ 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):

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

• (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.

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.