Symplectomorphisms of surfaces preserving a smooth function, I

Abstract

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty }$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also ${\mathcal{Z}}_{\omega }(f)\subset C^{\infty }(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and ${\mathcal{S}}_{\mathrm{id}}(f,\omega )$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $\omega$ and $f$. We construct a canonical map $\phi :{\mathcal{Z}}_{\omega }(f)\to {\mathcal{S}}_{\mathrm{id}}(f,\omega )$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. In particular, we obtain that ${\mathcal{S}}_{\mathrm{id}}(f,\omega )$ is either contractible or homotopy equivalent to the circle. Similar results hold in fact for a larger class of maps $M\to P$ whose singularities are equivalent to homogeneous polynomials without multiple factors.

13 pages, 3 figures, corrections in the abstract

References in corpus (2)

• Homotopy types of stabilizers and orbits of Morse functions on surfaces
• Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle

Cited by in corpus (1)

• Diffeomorphisms preserving Morse-Bott functions