Abstract:
The existence of fractional monodromy is proved for the compact
Lagrangian fibration on a symplectic 4-manifold that corresponds
to two oscillators with arbitrary non-trivial
resonant frequencies.
Here one means by the monodromy corresponding to
a loop in the total space of the fibration the
transformation of the fundamental group of a regular fibre,
which is diffeomorphic to the 2-torus.
In the example under consideration the fibration is defined by a
pair of functions
in involution, one of which is the Hamiltonian of the system of
two oscillators with frequency ratio
$m_1:(-m_2)$, where $m_1$, $m_2$ are arbitrary coprime
positive integers distinct from the trivial pair
$m_1=m_2=1$. This is a generalization of the result
on the existence of fractional monodromy in the case
$m_1=1$, $m_2=2$ published before.
Bibliography: 39 titles.
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