Geometry of Cantor Systems

Jiang, Yunping

doi:10.1090/S0002-9947-99-02214-X

Geometry of Cantor Systems
HTML articles powered by AMS MathViewer

by Yunping Jiang PDF

Trans. Amer. Math. Soc. 351 (1999), 1975-1987 Request permission

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.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
V. I. Arnol′d, Ordinary differential equations, The M.I.T. Press, Cambridge, Mass.-London, 1973. Translated from the Russian and edited by Richard A. Silverman. MR 0361233
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Charles Tresser and Pierre Coullet, Itérations d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A577–A580 (French, with English summary). MR 512110
Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 501179, DOI 10.1007/BF01020332
Mitchell J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys. 21 (1979), no. 6, 669–706. MR 555919, DOI 10.1007/BF01107909
Yun Ping Jiang, Geometry of geometrically finite one-dimensional maps, Comm. Math. Phys. 156 (1993), no. 3, 639–647. MR 1240589, DOI 10.1007/BF02096866
Y. Jiang, Generalized Ulam-von Neumann transformations, Thesis, Graduate School of CUNY (1990).
Y. Jiang, Nonlinearity, quasisymmetry, differentiability, and rigidity in one-dimensional dynamics, Preprint.
Yun Ping Jiang, Dynamics of certain nonconformal semigroups, Complex Variables Theory Appl. 22 (1993), no. 1-2, 27–34. MR 1277008, DOI 10.1080/17476939308814643
Oscar E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 427–434. MR 648529, DOI 10.1090/S0273-0979-1982-15008-X
Oscar E. Lanford III, A shorter proof of the existence of the Feigenbaum fixed point, Comm. Math. Phys. 96 (1984), no. 4, 521–538. MR 775044, DOI 10.1007/BF01212533
C. McMullen, Renormalization and $3$-manifolds which fiber over the circle, Preprint.
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
Steve Smale, The mathematics of time, Springer-Verlag, New York-Berlin, 1980. Essays on dynamical systems, economic processes, and related topics. MR 607330, DOI 10.1007/978-1-4613-8101-3
Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622