Hirokazu Shimauchi | Searcharxiv

arXiv2014

A remark on Zagier's observation of the Mandelbrot set

We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given. Bielefeld, Fisher and Haeseler presented Zagier's observation for the growth of denominator of the coefficients for the Mandelbrot set and Yamashita verified it in his master thesis. Our study takes into consideration both Zagier's observation and Yamashita's estimate.

math.DS↗

arXiv2014

Numerical solution of the Beltrami equation via a purely linear system

An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) in a planar disk. The disk is triangulated in a simple way and f is approximated by piecewise linear mappings; the images of the vertices of the triangles are defined by an overdetermined system of linear equations. (Certain apparently nonlinear conditions on the boundary are eliminated by means of a symmetry construction.) The linear system is sparse and its solution is obtained by standard least-squares, so the algorithm involves no evaluation of singular integrals nor any iterative procedure for obtaining a single approximation of f. Numerical examples are provided, including a deformation in a Teichmüller space of a Fuchsian group.

math.CV↗