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An example of limit of Lempert Functions
The Lempert function for several poles in a domain
of is defined at the point as the infimum of
over all the choices of points in the
unit disk so that one can find a holomorphic mapping from the disk to the
domain sending 0 to . This is always larger than the pluricomplex
Green function for the same set of poles, and in general different.
Here we look at the asymptotic behavior of the Lempert function for three
poles in the bidisk (the origin and one on each axis) as they all tend to the
origin. The limit of the Lempert functions (if it exists) exhibits the
following behavior: along all complex lines going through the origin, it
decreases like , except along three exceptional directions,
where it decreases like . The (possible) limit of the corresponding
Green functions is not known, and this gives an upper bound for it.Comment: 16 pages; references added to related work of the autho
Diophantine approximation on Veech surfaces
We show that Y. Cheung's general -continued fractions can be adapted to
give approximation by saddle connection vectors for any compact translation
surface. That is, we show the finiteness of his Minkowski constant for any
compact translation surface. Furthermore, we show that for a Veech surface in
standard form, each component of any saddle connection vector dominates its
conjugates. The saddle connection continued fractions then allow one to
recognize certain transcendental directions by their developments
Green vs. Lempert functions: a minimal example
The Lempert function for a set of poles in a domain of at a
point is obtained by taking a certain infimum over all analytic disks going
through the poles and the point , and majorizes the corresponding multi-pole
pluricomplex Green function. Coman proved that both coincide in the case of
sets of two poles in the unit ball. We give an example of a set of three poles
in the unit ball where this equality fails.Comment: v3: proof of the upper estimate for the Green function added;
accepted in Pacific Journal of Mathematic
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