Abstract
We study real solutions of a class of Painlevé VI equations. To each such solution we associate a geometric object, a one-parametric family of circular pentagons. We describe an algorithm that permits to compute the numbers of zeros, poles, 1-points and fixed points of the solution on the interval \({(1,+\infty)}\) and their mutual position. The monodromy of the associated linear equation and parameters of the Painlevé VI equation are easily recovered from the family of pentagons.
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Communicated by P. Deift
A. Eremenko: Supported by NSF Grant DMS-1665115.
A. Gabrielov: Supported by NSF Grant DMS-1161629.
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Eremenko, A., Gabrielov, A. Circular Pentagons and Real Solutions of Painlevé VI Equations. Commun. Math. Phys. 355, 51–95 (2017). https://doi.org/10.1007/s00220-017-2921-y
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DOI: https://doi.org/10.1007/s00220-017-2921-y