A Mandelbrot set whose boundary is piecewise smooth

Barnsley, M. F.; Hardin, D. P.

doi:10.1090/S0002-9947-1989-1011232-6

A Mandelbrot set whose boundary is piecewise smooth
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by M. F. Barnsley and D. P. Hardin PDF

Trans. Amer. Math. Soc. 315 (1989), 641-659 Request permission

Abstract:

It is proved that the Mandelbrot set associated with the pair of maps ${w_{1,2}}:{\mathbf {C}} \to {\mathbf {C}}, {w_1}(z) = sz + 1, {w_2}(z) = {s^\ast }z - 1$, with parameter $s \in {\mathbf {C}}$, is connected and has piecewise smooth boundary.

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On the existence and non-existence of natural boundaries for non-intregrable dynamical systems