Hofer's distance on diameters and the Maslov index (Q2909339)

This paper is inspired by the paper [\textit{M. Khanevsky}, J. Topol. Anal. 1, No. 4, 407--416 (2009; Zbl 1205.53082)]. The author considers the open unit disk \(D\) on the plane \(\mathbb{R}^2\) centered at the origin with the symplectic structure \(\omega = \frac{1}{\pi} dx \wedge dy\) and a time-dependent Hamiltonian \(H\). A diameter is a curve that is isotopic to the standard diameter \(L_0 = D \cap (\mathbb{R} \times \left\{ 0 \right\})\) via a compactly supported Hamiltonian isotopy. The space \(\mathcal{E}\) of all diameters is endowed with the Hofer distance: NEWLINE\[NEWLINE d(L_1, L_2) = \inf \int_0^1 \left(\max\limits_{x \in D} H_t - \min\limits_{x \in D} H_t\right) \, dt, NEWLINE\]NEWLINE where the infimum is taken over all smooth and compactly supported time-dependent Hamiltonians \(H_t\) such that \(\phi^t_H(L_1) = L_2\) and \(\phi^t_H\) is the Hamiltonian isotopy generated by \(H\) (about the properties of this metric see [\textit{Yu. V. Chekanov}, Math. Z. 234, No. 3, 605--619 (2000; .Zbl 0985.37052)]).NEWLINENEWLINEIf \(L\) is a diameter transverse to the diameter \(L_0\), and \(\{x_i\}_{i = 1, \dots, N}\) is the set of intersection points, then \(\mu_{\min}(L) =\min\limits_{i = 1, \dots, N} \mu(x_i)\), \(\mu_{\max}(L) = \max\limits_{i = 1, \dots, N} \mu(x_i)\), where \(\mu(x_i)\) is the Maslov index of the corresponding intersection point. The main result of the paper is Theorem 1.2 which states that, for any diameter \(L\) transverse to \(L_0\), with at least two transverse intersection points with \(L_0\), NEWLINE\[NEWLINE d(L, L_0) \leq \mu_{\max}(L) - \mu_{\min}(L) - \frac{1}{2}. NEWLINE\]NEWLINE This result complements the results of Khanevsky's paper, one of which says that \(d\) is dominated by the number of intersection points.