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On the Borderline of Real and Complex Dynamics

In: Proceedings of the International Congress of Mathematicians

Author

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  • Mikhail Lyubich

    (State Univ. of New York, Mathematics Department and Institute for Math. Sciences)

Abstract

We will describe recent developments in several intimately related problems of complex and real one-dimensional dynamics: rigidity of polynomials and local connectivity of the Mandelbrot set, measure of Julia sets, and attractors of quasi-quadratic maps. A combinatorial basis for this study is provided by the Yoccoz puzzle. The main problem is to understand the geometry of the puzzle. Our main geometric result is that in the quadratic case its principal moduli grow linearly. Renormalization ideas are strongly involved in the discussion. The interplay between real and complex dynamics enlightens both. In the end we will brieftly discuss a new geometric object which can be associated to a rational function, a hyperbolic orbifold 3-lamination.

Suggested Citation

  • Mikhail Lyubich, 1995. "On the Borderline of Real and Complex Dynamics," Springer Books, in: S. D. Chatterji (ed.), Proceedings of the International Congress of Mathematicians, pages 1203-1215, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9078-6_114
    DOI: 10.1007/978-3-0348-9078-6_114
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