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The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. (English) Zbl 0715.58018

The paper deals with the dynamical systems which are obtained by iterating rational maps of the Riemann sphere. Sullivan’s definition of the mapping class groups (MCG) for rational maps is recalled and the representation of MCG(R) as a group of automorphisms of the shift is studied. The two parameter family of quadratic rational maps \(R(z;\lambda,b)=1/\lambda (z+b+1/z)\) is investigated from the point of view of the properties of its Julia set and Mandelbrot set. The restriction of \(R_{\lambda,b}\) to a suitable neighborhood of \(\infty\) is conjugated to a linear map. The authors construct the mapping making this conjugacy.
Reviewer: A.Klíč

MSC:

37B99 Topological dynamics
37E99 Low-dimensional dynamical systems
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30D45 Normal functions of one complex variable, normal families
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References:

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