Semi-local Liouville equivalence of complex Hamiltonian systems defined by rational Hamiltonian

Consider the holomorphic function $f = az^2 + R(w)$, where $z$ and $w$ are complex variables and $R$ is a rational function. Let $D_{\xi_0, \varepsilon}$ be a small disc around $\xi_0 \in \mathbb C$. Function $f$ defines the foliation in the neighborhood $f^{-1}(D_{\xi_0, \varepsilon})$ of the (singular) fiber $f^{-1}(\xi_0)$. We give a complete topological classification of such foliations

On monodromy in integrable Hamiltonian systems

In the context of integrable Hamiltonian systems, the notion of monodromy goes back to Duistermaat, who defined this invariant as the first obstruction to the existence of good global coordinates in such systems; the coordinates are known as global action-angle coordinates in the literature. Since then the notion of monodromy has received considerable interest and has also been generalized in several different directions.In this PhD thesis we give a systematic study of monodromy invariants in integrable Hamiltonian systems. We mainly study the following three different types of such invariants: Hamiltonian, fractional, and scattering monodromy. We provide new general methods which allow one to compute these invariants in many concrete examples of integrable systems and establish new connections to well-known mathematical theories, including Morse theory, Seifert manifolds, and scattering theory.The present thesis consists of 5 chapters. In Chapters 1-3 we study monodromy in compact integrable systems, for which the motion generically takes place on invariant tori. In Chapters 4 and 5 we consider the case of scattering systems, where the motion is unbounded. Throughout this thesis, applications of the obtained theoretical results to concrete examples of integrable Hamiltonian systems are discussed

An obstruction to Delaunay triangulations in Riemannian manifolds

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on $P$ are required. A natural one is to assume that $P$ is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2.Comment: This is a revision and extension of a note that appeared as an appendix in the (otherwise unpublished) report arXiv:1303.649

Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

We show that every parabolic orbit of a two-degree of freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the integrals of motion is Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets

Recent advances in the monodromy theory of integrable Hamiltonian systems

The notion of monodromy was introduced by J. J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalisations. In particular, we will discuss the monodromy around a focus-focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems

Towards Hypersemitoric Systems

This survey gives a short and comprehensive introduction to a class of finite-dimensional integrable systems known as hypersemitoric systems, recently introduced by Hohloch and Palmer in connection with the solution of the problem how to extend Hamiltonian circle actions on symplectic 4-manifolds to integrable systems with `nice' singularities. The quadratic spherical pendulum, the Euler and Lagrange tops (for generic values of the Casimirs), coupled-angular momenta, and the coupled spin oscillator system are all examples of hypersemitoric systems. Hypersemitoric systems are a natural generalization of so-called semitoric systems (introduced by Vu Ngoc) which in turn generalize toric systems. Speaking in terms of bifurcations, semitoric systems are `toric systems with/after supercritical Hamiltonian-Hopf bifurcations'. Hypersemitoric systems are `semitoric systems with, among others, subcritical Hamiltonian-Hopf bifurcations'. Whereas the symplectic geometry and spectral theory of toric and semitoric sytems is by now very well developed, the theory of hypersemitoric systems is still forming its shape. This short survey introduces the reader to this developing theory by presenting the necessary notions and results as well as its connections to other areas of mathematics and mathematical physics.Comment: 26 pages, 8 figure