Classification of hyperbolic singularities of rank zero of integrable Hamiltonian systems

A complete invariant is constructed that is a solution of the problem of semilocal classification of saddle singularities of integrable Hamiltonian systems. Namely, a certain combinatorial object (an -graph) is associated with every nondegenerate saddle singularity of rank zero; as a result, the problem of semilocal classification of saddle singularities of rank zero is reduced to the problem of enumeration of the -graphs. This enables us to describe a simple algorithm for obtaining the lists of saddle singularities of rank zero for a given number of degrees of freedom and a given complexity. Bibliography: 24 titles.

[1]  A. Oshemkov.  Saddle singularities of complexity 1 of integrable Hamiltonian systems , 2011 .

[2]  Alexey V. Bolsinov,et al.  Integrable Hamiltonian Systems: Geometry, Topology, Classification , 2004 .

[3]  N. Leung,et al.  Almost toric symplectic four-manifolds , 2003, math/0312165.

[4]  Margaret Symington.  Four dimensions from two in symplectic topology , 2002, math/0210033.

[5]  N. Zung.  Another Note on Focus-Focus Singularities , 2001, math/0110148.

[6]  N. T. Zung,et al.  Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities , 2001, math/0106013.

[7]  N. T. Zung,et al.  A note on focus-focus singularities , 1997, math/0110147.

[8]  V. Matveev.  Integrable Hamiltonian system with two degrees of freedom. The topological structure of saturated neighbourhoods of points of focus-focus and saddle-saddle type , 1996 .

[9]  L. Lerman,et al.  Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $ mathbb R^2$ in extended neighbourhoods of simple singular points. Iii. Realization , 1994 .

[10]  Maorong Zou.  Monodromy in two degrees of freedom integrable systems , 1992 .

[11]  A. Fomenko.  A BORDISM THEORY FOR INTEGRABLE NONDEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. A NEW TOPOLOGICAL INVARIANT OF HIGHER-DIMENSIONAL INTEGRABLE SYSTEMS , 1992 .

[12]  A. Bolsinov.  Methods of calculation of the Fomenko-Zieschang invariant , 1991 .

[13]  A. Fomenko,et al.  A TOPOLOGICAL INVARIANT AND A CRITERION FOR THE EQUIVALENCE OF INTEGRABLE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM , 1991 .

[14]  L. Eliasson.  Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case , 1990 .

[15]  A. Fomenko,et al.  THE TOPOLOGY OF INTEGRAL SUBMANIFOLDS OF COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS , 1989 .

[16]  A. Fomenko.  The Topology of Surfaces of Constant Energy in Integrable Hamiltonian Systems, and Obstructions to Integrability , 1987 .

[17]  A. Bolsinov,et al.  Singularities of integrable Hamiltonian systems , 2006 .

[18]  Владимир Сергеевич Матвеев,et al.  Интегрируемые гамильтоновы системы с двумя степенями свободы. Топологическое строение насыщенных окрестностей точек типа фокус-фокус и седло-седло@@@Integrable Hamiltonian system with two degrees of freedom. The topological structure of saturated neighbourhoods of points of focus-focus and saddle-sa , 1996 .

[19]  Percy Deift,et al.  Integrable Hamiltonian systems , 1996 .

[20]  A. Fomenko.  Topological invariants of Liouville integrable Hamiltonian systems , 1988 .