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Real Bounds in Complex Dynamics

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Real and Complex Dynamical Systems

Part of the book series: NATO ASI Series ((ASIC,volume 464))

Abstract

In these lectures we shall review some results in complex dynamics in which real bounds play a role. Then we shall give a survey on these real bounds and explain how to apply these.

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References

  1. L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York (1979).

    MATH  Google Scholar 

  2. L.V. Ahlfors, Lectures on quasiconformal mappings,Van Nostrand, Princeton, N.J. (1966), Reprinted in 1987 by Wadsworth and Brooks/Cole.

    Google Scholar 

  3. B. Branner and J.H.Hubbard, The iteration of cubic polynomials I, Acta Math. 160, (1988), 143–206, The iteration of cubic polynomials II, Acta Math. 169, (1992), 229–325.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Bruin, G. Keller, T. Nowicki and S. van Strien, Absorbing Cantor sets in dynamical systems: Fibonacci maps,Stonybrook IMS preprint 1994/2.

    Google Scholar 

  5. M. R. Herman, Conjugaison quasis symmetrique des homeomorphismes analytiques du cercle a des rotations,preliminary manuscript.

    Google Scholar 

  6. M.V. Jacobson and G. Swiatek, Quasisymmetric conjugacies between unimodal maps, Stonybrook IMS Preprint 1991/16.

    Google Scholar 

  7. G. Keller and T. Nowicki, Fibonacci maps re(ae)visited, Preprint (1992).

    Google Scholar 

  8. R. de la Llave, in preparation.

    Google Scholar 

  9. G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials, Preprint (1995).

    Google Scholar 

  10. M.Yu. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial. Stonybrook IMS Preprint 1991/10.

    Google Scholar 

  11. M.Yu. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps. Stonybrook IMS Preprint 1992/18.

    Google Scholar 

  12. M. Lyubich and J. Milnor, The unimodal Fibonacci map, preprint StonyBrook (1991).

    Google Scholar 

  13. C. McMullen, Complex dynamics and renormalization, Preprint Berkeley (1993).

    Google Scholar 

  14. W. de Melo and S. van Strien (1988): One-dimensional dynamics: the Schwarzian derivative and beyond. Bull. A.M.S. 18 159–162.

    Google Scholar 

  15. W. de Melo and S. van Strien (1989): A structure theorem in one-dimensional dynamics. Ann. Math. 129 519–546.

    Google Scholar 

  16. W. de Melo and S. van Strien: One-Dimensional Dynamics. Ergebnisse der Mathematik, 3. Folge, Volume 25, Springer-Verlag, Berlin etc. June 1993.

    Google Scholar 

  17. C. Petersen, Local connectivity of some Julia sets containing a circle with an irrational rotation, Preprint I.H.E.S. 1994.

    Google Scholar 

  18. M. Shishikura, unpublished.

    Google Scholar 

  19. D. Sullivan: Differentiable structures on fractal like sets, determined by intrinsic scaling functions on dual Cantor sets. In: “Nonlinear Evolution and Chaotic Phenomena” Plenum, New York, (1988).

    Google Scholar 

  20. D. Sullivan: The universalities of Milnor, Feigenbaum and Bers. To appear in “Topological Methods in Modern Mathematics”, Proceedings of the Symposium held in honor of John Milnor’s 60th birthday, SUNY at Stony Brook, June 14–21, 1991.

    Google Scholar 

  21. D. Sullivan: Bounds, quadratic differentials, and renormalization conjectures. In: “A.M.S. Centennial Publications, vol. 2, Mathematics into the Twenty-first Century (1988)”.

    Google Scholar 

  22. G. Swigtek (1988): Rational rotation numbers for maps of the circle. Communication in Math. Phys. 119, 109–128.

    Google Scholar 

  23. G. Swigtek, Hyperbolicity is dense in the real quadratic family. Stonybrook IMS Preprint 1992/10.

    Google Scholar 

  24. F. Tangerman and P. Veerman, Scaling in cirlce maps I. Preprint I, SUNY, Stony Brook, 1990/10.

    Google Scholar 

  25. J.-C. Yoccoz (1984b): Il n’y a pas de contre-example de Denjoy analytique. C.R. Acad. Sci. Paris, 298, série I, 141–144.

    Google Scholar 

  26. J.-C. Yoccoz (1991): On the local connectivity of the Mandelbrot set. To appear.

    Google Scholar 

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© 1995 Springer Science+Business Media Dordrecht

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van Strien, S. (1995). Real Bounds in Complex Dynamics. In: Branner, B., Hjorth, P. (eds) Real and Complex Dynamical Systems. NATO ASI Series, vol 464. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8439-5_9

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  • DOI: https://doi.org/10.1007/978-94-015-8439-5_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4565-2

  • Online ISBN: 978-94-015-8439-5

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