An Arnold-type principle for non-smooth objects (Q2124780)

If \((M,\omega)\) is a closed and connected symplectic manifold, then a Hamiltonian diffeomorphism is a diffeomorphism which arises as the time-one map of a Hamiltonian flow \(\varphi^H_t\) of a smooth Hamiltonian \(H:\mathbb{R}/\mathbb{Z}\to\mathbb{R}\). The set of all Hamiltonian diffeomorphisms is denoted by \(\mathrm{Ham}(M,\omega)\). There exists a one-to-one correspondence between the set of fixed points of \(\varphi^H_1\) and that of contractible \(1\)-periodic solutions of the Hamiltonian equation \(\dot{x}(t)=X_H(t, x(t))\), where \(\omega(X_H(t),\cdot)=-dH_t\). The homological Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold \((M,\omega)\) must have at least as many fixed points as the cup-length \(\mathrm{cl}(M)\) of \(M\). When \([\omega]|\pi_2(M)=0\), this conjecture was proved by \textit{A. Floer} [Commun. Pure Appl. Math. 42, No. 4, 335--356 (1989; Zbl 0683.58017)] and \textit{H. Hofer} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, No. 5, 465--499 (1988; Zbl 0669.58006)]. \textit{Lê Hông Vân} and \textit{K. Ono} [in: Contact and symplectic geometry. Cambridge: Cambridge University Press. 268--295 (1996; Zbl 0874.53030)] also proved this conjecture for negatively monotone symplectic manifolds. It is natural to ask if in the above results the smoothness of Hamiltonian diffeomorphisms or Hamiltonian functions could be weakened in some sense. Suppose that a Hamiltonian function \(H:\mathbb{R}/\mathbb{Z}\to\mathbb{R}\) is only \(C^1\). The Hamiltonian equation \(\dot{x}(t)=X_H(t, x(t))\) cannot yield a flow and thus a Hamiltonian diffeomorphism in general. If \(\omega|_{\pi_2(M)}=0\), Lu showed that the equation \(\dot{x}(t)=X_H(t, x(t))\) must have at least \(\mathrm{cl}(M)\) contractible \(1\)-periodic solutions [\textit{G. Lu}, Bull. Inst. Math., Acad. Sin. 33, No. 1, 69--76 (2005; Zbl 1075.37021)]. Recall that a homeomorphism on \((M,\omega)\) is said to be a Hamiltonian homeomorphism if it is the \(C^0\)-limit of a sequence of Hamiltonian diffeomorphisms. Matsumoto proved that Hamiltonian homeomorphisms of surfaces satisfy the above Arnold conjecture [\textit{S. Matsumoto}, Topology Appl. 104, No. 1--3, 191--214 (2000; Zbl 0974.37040)]. However, in striking contrast to this, Buhovsky, Humiliére and Seyfaddini recently showed that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point [\textit{L. Buhovsky} et al., Invent. Math. 213, No. 2, 759--809 (2018; Zbl 1395.37037)]. On the other hand, when \(\omega|_{\pi_2(M)}=0=c_1|_{\pi_2(M)}\) and the total number of spectral invariants of a Hamiltonian homeomorphism \(\phi\) on \((M, \omega)\) is smaller than \(\mathrm{cl}(M)\), Buhovsky, Humiliére and Seyfaddini demonstrated that the set of fixed points of \(\phi\) is homologically non-trivial, hence is infinite [\textit{L. Buhovsky} et al., Math. Ann. 380, No. 1--2, 293--316 (2021; Zbl 1471.53067)]. This generalized the above Arnold conjecture by [\textit{W. Howard}, ``Action selectors and the fixed point set of a Hamiltonian diffeomorphism'', Preprint, \url{arXiv:1211.0580}; \textit{V. Humilière} et al., Ann. Sci. Éc. Norm. Supér. (4) 49, No. 3, 633--668 (2016; Zbl 1341.53114)]. In the paper under review, similar results are proved in two new settings. \begin{itemize} \item[(i)] Let \(O_N\) be the zero section of the cotangent bundle \(T^\ast N\) of a closed and connected manifold \(N\) and let \(\phi\) denote a compactly supported Hamiltonian homeomorphism of \(T^\ast N\). If the number of spectral invariants of \(L\) is smaller than \(\mathrm{cl}(N)\), it is claimed in Theorem~1.1 that \(L\cap O_N\) is homologically non-trivial, hence it is infinite. When \(\dim N=2\) it is also proved in Proposition 1.2 that there is a Hamiltonian homeomorphism \(\psi\) of \(T^\ast N\) such that the \(C^0\)-Lagrangian \(L=\psi(O_N)\) has only one intersection with the zero-section \(O_N\). \item[(ii)] Let \(L_i\) be a sequence of Legendrian submanifolds in the \(1\)-jet bundle \(J^1N\) which are contact isotopic to the zero-section \(O_N\times\{0\}\). Suppose that this sequence has a compact Hausdorff limit \(L\subset J^1N\) whose spectrum \(\mathrm{Spec}(L)\) has cardinality strictly less than \(\mathrm{cl}(N)\). Then there exists \(\lambda\in \mathrm{Spec}(L)\) such that \(L\cap (O_N\times\{\lambda\})\) is homologically non-trivial in \(O_N\times\{\lambda\}\), and thus \(L\cap (O_N\times\mathbb{R})\) is infinite (Theorem~1.5). \end{itemize}