Abstract
We construct a Markov partition for a Feigenbaum-like mapping. We prove that this Markov partition has bounded nearby geometry property similar to that for a geometrically finite one-dimensional mappings [8]. Using this property, we give a simple proof that any two Feigenbaum-like mappings are topologically conjugate and the conjugacy is quasisymmetric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors, L.: Lectures on Quasiconformal Mappings. Princeton, NJ: Van Nostrand-Reinhold, 1966
Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, Vol.470, Berlin, Heidelberg, New York: Springer, 1975
Coullet, P., Tresser, C.: Itération d'endomorphismes et groupe de renormalisation. C.R. Acad. Sci. Paris Ser., A-B287, A577-A580 (1978)
Feigenbaum, M.: Quantitative universality for a class of non-linear transformation. J. Stat. Phys.19, 25–52 (1978)
Feigenbaum, M.: The universal metric properties of non-linear transformation. J. Stat. Phys21, 669–706 (1979)
Guckenheimer, J.: Limit sets ofS-unimodal maps with zero entropy. Commun. Math. Phys.110, 655–659 (1987)
Guckenheimer, J.: Sensitive dependence on initial conditions for one-dimensional maps. Commun. Math. Phys.70, 133–160 (1979)
Jiang, Y.: Geometry of geometrically finite one-dimensional mappings. Commun. Math. Phys.156, 639–647 (1993)
Jiang, Y.: Infinitely renormalizable quadratic Julia sets. Research Report of FIM, ETH, Zürich, June, 1994
Jiang, Y.: On quasisymmetrical classification of infinitely renormalizable maps — I. Maps with Feigenbaum topology. and II. Remarks on maps with a bounded type topology. IMS preprint series 1991/199, SUNY at Stony Brook, 1991
Lyubich, M.: Geometry of quadratic polynomials: Moduli, rigidity, and local connectivity. IMS preprint series, 1993/9, SUNY@Stony Brook
de Melo, W., van Strien, S.: One-dimensional Dynamics. Berlin, Heidelberg, New York: Springer, 1993
de Melo, W., van Strien, S.: A structure theorem in one dimensional dynamics. Ann. Math.129, 519–546 (1989)
Milnor, J., Thurston, W.: On iterated maps of the interval: I and II. In: Lecture Notes in Mathematics., Vol.1342, Berlin, Heidelberg, New York: Springer, 1988, pp. 465–563
Paluba, W.: Talks in Graduate Center of CUNY, 1991 and 1992
Sinai, Ya.: Markov partitions andC-diffeomorphisms. Funct. Anal. and its Appl.2, no. 1, 64–89 (1968)
Swiatek, G.: Hyperbolicity is dense in real quadratic family. IMS preprint series 1992/9, SUNY@StonyBrook, 1992
Sullivan, D.: Bounds, quadratic differentials, and renormalization conjectures. American Mathematical Society Centennial Publications, Volume2: Mathematics into the Twenty-first Century, Providence, RI: AMS, 1991
Author information
Authors and Affiliations
Additional information
Communicated by J.-P. Eckmann
Partially supported by a PSC-CUNY and NSF grants
Rights and permissions
About this article
Cite this article
Jiang, Y. Markov partitions and Feigenbaum-like mappings. Commun.Math. Phys. 171, 351–363 (1995). https://doi.org/10.1007/BF02099274
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099274