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Dulac maps of real saddle-nodes
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Abstract
Consider a germ of a holomorphic vector field at the origin on the coordinate complex plane. This germ is called a saddle-node if the origin is its singular point, one of its eigenvalues at zero is zero, and the other is not. A saddle-node germ is real if its restriction to the real plane is real. The monodromy transformation for this germ has a multiplier at zero equal to 1. The germ of this map is parabolic and admits a ‘normalizing cochain’. In this note we express the Dulac map of any real saddle-node up to a left composition with a real germ
(
C
,
0
)
→
(
C
,
0
)
through one component of the cochain normalizing the monodromy transformation.
Title: Dulac maps of real saddle-nodes
Description:
Abstract
Consider a germ of a holomorphic vector field at the origin on the coordinate complex plane.
This germ is called a saddle-node if the origin is its singular point, one of its eigenvalues at zero is zero, and the other is not.
A saddle-node germ is real if its restriction to the real plane is real.
The monodromy transformation for this germ has a multiplier at zero equal to 1.
The germ of this map is parabolic and admits a ‘normalizing cochain’.
In this note we express the Dulac map of any real saddle-node up to a left composition with a real germ
(
C
,
0
)
→
(
C
,
0
)
through one component of the cochain normalizing the monodromy transformation.
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