11,997 research outputs found

    An example of limit of Lempert Functions

    Full text link
    The Lempert function for several poles a0,...,aNa_0, ..., a_N in a domain Ω\Omega of Cn\mathbb C^n is defined at the point zΩz \in \Omega as the infimum of j=0Nlogζj\sum^N_{j=0} \log|\zeta_j| over all the choices of points ζj\zeta_j in the unit disk so that one can find a holomorphic mapping from the disk to the domain Ω\Omega sending 0 to zz. 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 (3/2)logz(3/2) \log |z|, except along three exceptional directions, where it decreases like 2logz2 \log |z|. 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

    Get PDF
    We show that Y. Cheung's general ZZ-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

    Full text link
    The Lempert function for a set of poles in a domain of Cn\mathbb C^n at a point zz is obtained by taking a certain infimum over all analytic disks going through the poles and the point zz, 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
    corecore