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Dynamics of analytic endomorphisms on the sphere. (Dynamik analytischer Endomorphismen auf der Sphäre.) (German) Zbl 0855.30021
In the past, several surveys on the dynamics of holomorphic endomorphisms of the complex sphere have been appeared but these have not covered the measure theoretic aspects in this field. The authors of the present survey are the first focusing on invariant measures on Julia sets and related topics.
After a brief motivation the authors recall definitions as well as basic properties of Julia and Fatou sets including Sullivan’s non-wandering theorem and the classification theorem for periodic Fatou components.
In the second chapter, the authors present a detailed discussion of the dynamics on invariant Fatou components. Starting with universal coverings and the Decktransformation groups they sketch the proofs of the non-wandering theorem and the classification theorem.
All important is the third chapter. Here, the authors carefully explain measure theoretic aspects of iteration theory.
The last chapter is devoted to Makarov’s theorem on harmonic measures on the boundary of simply connected domains. In particular, the authors show how to improve this result by using the concepts presented in the previous sections.
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics
26A18 Iteration of real functions in one variable
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics