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Pasting together Julia sets: a worked out example of mating. (English) Zbl 1115.37051
Summary: The operation of “mating” two suitable complex polynomial maps \(f_1\) and \(f_2\) constructs a new dynamical system by carefully pasting together the boundaries of their filled Julia sets so as to obtain a copy of the Riemann sphere, together with a rational map \(f_1*f_2\) from this sphere to itself. This construction is particularly hard to visualize when the filled Julia sets \(K(f_i)\) are dendrites, with no interior. This note will work out an explicit example of this type, with effectively computable maps from \(K(f_1)\) and \(K(f_2)\) onto the Riemann sphere.

MSC:
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
28A80 Fractals
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