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Register a new userVu Ngocs Conjecture on focus-focus singular fibers with multiple pinched points
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We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(mathbb{Z}_2 times D_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields $X_{f_1}, X_{f_2}$ associated to the components $f_1, f_2 colon (M, omega) to mathbb{R}$ of the integrable system on the symplectic manifold $(M,omega)$ via the differential equation $omega(X_{f_i}, cdot) = mathop{}!mathrm{d} f_i$, near the singular fiber containing the $k$ focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.
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About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $mathbb{S}^2 times mathbb{R}^2$ and coupled angular momenta on $mathbb{S}^2 times mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $mathbb{S}^2 times mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently big then the local-global convexity principle breaks down, and the base spaces can be globally non-convex even for compact manifolds. As one of surprising examples, we construct a 2-dimensional integral affine black hole, which is locally convex but for which a straight ray from the center can never escape.
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