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On the dynamics of polynomial-like mappings. (English) Zbl 0587.30028
The standard Mandelbrot set is defined as follows. Let \(P(c,z)=z^ 2+c_ 2\), \(c\in {\mathbb{C}}\), and denote by \(P^ n(c,z)\) the n-th iterate of the map \(z\to z^ 2+c\). Then \(M=\{c:\) \(P^ n(c,0)\) is bounded\(\}\). The characteristic shape M appears in many diagrams relating to the dynamical behaviour of other one parameter families f(z,c). In particular M contains many small copies of itself. Also, if we try to solve \[ f(\lambda,z)\equiv (z-1)(z++\lambda)(z+-\lambda)=0 \] by Newton’s method with initial approximation \(z=0\) and mark those parameter values \(\lambda\in {\mathbb{C}}\) for which the process does not converge to a root of \(f(\lambda,z)=0\), small copies of M appear. Newton’s method is equivalent to the iteration of \(z\to N(\lambda,z)=z-f(\lambda,z)/(f'(\lambda,z),\) starting at \(z=0\). For certain \(\lambda_ 0,z_ 0\), \(k\in {\mathbb{N}}\), one has \(N^ k(\lambda_ 0,0)=0\) and then \(N^ k(\lambda_ 0,z)\sim z^ 2\) near 0. For \(\lambda\) near \(\lambda_ 0\) the map \(N^ k(\lambda,z)\) ”behaves like” \(z+c(\lambda)\) for some c(\(\lambda)\). It turns out that the map \(\lambda\) \(\to c(\lambda)\) is a homeomorphism and \(c^{-1}(M)\) is a neighbourhood of \(\lambda_ 0\) such that for \(\lambda \in c^{- 1}(M)\), the sequence \(N^{kn}(\lambda_ 0)\) stays in \(c^{-1}(M)\) and so does not converge to one of the roots of f.
The rigorous theory to justify the above description and extend it to more general cases rests on the authors’ idea of polynomial-like map of degree d, i.e. a triple (U,U’,f) where U and U’ are open subsets of \({\mathbb{C}}\) isomorphic to discs, with U’ relatively compact in U and f: U’\(\to U\) an analytic map, proper of degree d. The straightening theorem says that every such polynomial-like mapping can be related by a quasiconformal map to a polynomial of degree d. Using these ideas the authors build up a theory which, as special applications, explains the examples mentioned above.
Reviewer: I.N.Baker

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics
37G99 Local and nonlocal bifurcation theory for dynamical systems
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