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Geometry of Cantor Systems


Author: Yunping Jiang
Journal: Trans. Amer. Math. Soc. 351 (1999), 1975-1987
MSC (1991): Primary 57F25, 58F11
DOI: https://doi.org/10.1090/S0002-9947-99-02214-X
Published electronically: January 27, 1999
MathSciNet review: 1475687
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Abstract: A Cantor system is defined. The geometry of a certain family of Cantor systems is studied. Such a family arises in dynamical systems as hyperbolicity is created. We prove that the bridge geometry of a Cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap.


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  • [AH] L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand (1966). MR 34:336
  • [AR] V. I. Arnold, Ordinary Differential Equations, M.I.T. Press: Cambridge, MA (1973 (Russian original, Moscow, 1971)). MR 50:13679
  • [BI] L. Bieberbach, Conformal Mapping, Chelsea Publishing Company, New York, 1953. MR 14:462c
  • [CT] P. Coullet and C. Tresser, Itération d'endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Ser., A-B 287 (1978), A577-A580. MR 80b:58043
  • [CE] P. Collet and P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics 1 (1980). MR 82j:58078
  • [FE1] M. Feigenbaum, Quantitative universality for a class of non-linear transformations, J. Stat. Phys. 19 (1978), 25-52. MR 58:18601
  • [FE2] M. Feigenbaum, The universal metric properties of non-linear transformations, J. Stat. Phys. 21 (1979), 669-706. MR 82e:58072
  • [JI1] Y. Jiang, Geometry of geometrically finite one-dimensional maps, Commun. in Math. Phys. 156 (1993), 639-647. MR 95f:58033
  • [JI2] Y. Jiang, Generalized Ulam-von Neumann transformations, Thesis, Graduate School of CUNY (1990).
  • [JI3] Y. Jiang, Nonlinearity, quasisymmetry, differentiability, and rigidity in one-dimensional dynamics, Preprint.
  • [JI4] Y. Jiang, Dynamics of certain non-conformal semi-groups, Complex Variables 22 (1993), 27-34. MR 95g:30025
  • [LA1] O. E. Lanford III, A computer-assistant proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. 6 (1982), 427-434. MR 83g:58051
  • [LA2] O. E. Lanford III, A shorter proof of the existence of Feigenbaum fixed point, Commun. in Math. Phys. 96 (1984), 521-538. MR 86c:58121
  • [MC] C. McMullen, Renormalization and $3$-manifolds which fiber over the circle, Preprint.
  • [MV] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, Heidelberg, 1993. MR 95a:58035
  • [SM] S. Smale, The Mathematics of Time: Essays on Dynamical Systems, Economic Processes and Related Topics, Springer-Verlag, New York, Heidelberg, Berlin, 1980. MR 83a:01068
  • [SU] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, Mathematics into the Twenty-First Century 2 (1992). MR 93k:58194

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Additional Information

Yunping Jiang
Affiliation: Department of Mathematics, Queens College, The City University of New York, Flushing, New York 11367-1597 and Department of Mathematics, Graduate School of The City University of New York, New York, New York 10036
Email: yunqc@yunping.math.qc.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02214-X
Received by editor(s): February 12, 1996
Received by editor(s) in revised form: December 2, 1996
Published electronically: January 27, 1999
Additional Notes: Partially supported by an NSF grant and PSC-CUNY awards
Article copyright: © Copyright 1999 American Mathematical Society

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