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Attracting dynamics of exponential maps. (English) Zbl 1088.30016
Author’s abstract:
We give a complete classification of hyperbolic components in the space of iterated exponential maps $$z\mapsto\lambda\exp(z)$$, and we describe a preferred parametrization of those components. More precisely, we associate to every hyperbolic component of period $$n$$ a finite symbolic sequence of length $$n- 1$$, we show that every such sequence is realized by a hyperbolic component, and the hyperbolic component specified by any such sequence is unique. This leads to a complete classification of all exponential maps with attracting dynamics, which is a fundamental step in the understanding of exponential parameter space.

MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 33B10 Exponential and trigonometric functions 37B10 Symbolic dynamics 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 37C20 Generic properties, structural stability of dynamical systems 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F20 Combinatorics and topology in relation with holomorphic dynamical systems 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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