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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