Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Smooth classification of geometrically finite one-dimensional maps


Author: Yunping Jiang
Journal: Trans. Amer. Math. Soc. 348 (1996), 2391-2412
MSC (1991): Primary 58F03, 58F19, 58F34, 30F35
DOI: https://doi.org/10.1090/S0002-9947-96-01487-0
MathSciNet review: 1321579
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Hölder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete $C^{1}$-invariants within a mixing topological conjugacy class.


References [Enhancements On Off] (What's this?)

  • 1. L. V. Ahlfors, Lectures on Quasiconformal Maps, Van Nostrand Company, Princeton, New Jersey, 1966. MR 34:336
  • 2. M. Feigenbaum, Presentation functions, fixed points and a theory of scaling function dynamics, J. Statist. Phys. 52 (1988), 527--569. MR 90a:58112
  • 3. M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I. H. E. S., No. 49 (1979), 5-233. MR 81h:58039
  • 4. Y. Jiang, Generalized Ulam-von Neumann transformations, Thesis, 1990, the Graduate School of CUNY.
  • 5. Y. Jiang, On Ulam-von Neumann transformations, Commun. Math. Phys. 172 (1995), 449--459.
  • 6. Y. Jiang, Dynamics of certain smooth one-dimensional mappings -- I. The $C^{1+\alpha }$-Denjoy-Koebe distortion lemma, IMS preprint series 1991/1, SUNY at Stony Brook.
  • 7. Y. Jiang, Geometry of geometrically finite one-dimensional maps, Commun. in Math. Phys. 156, 639-647, 1993. MR 95f:58033
  • 8. R. de la Llave and R. Moriyón, Invariant for smooth conjugacy of hyperbolic dynamical systems. IV, Comm. in Math. Phys. 109, 369-378, 1987. MR 90h:58064
  • 9. J. Milnor and W. Thurston, On iterated maps of the interval I and II, Preprint, Princeton University, 1977; finally published in Lecture Notes in Math. 1342 (1988), 465--563. MR 90a:58083
  • 10. M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergod. Th & Dynam. Sys., 5, 1987, 285-289. MR 87g:58104
  • 11. D. Sullivan, Private conversation.
  • 12. D. Sullivan, Differentiable structure on fractal-like sets determined by intrinsic scaling functions on dual Cantor sets, Proceedings of Symposia in Pure Mathematics, Vol. 48, 1988, 15--23. MR 90k:58141

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F03, 58F19, 58F34, 30F35

Retrieve articles in all journals with MSC (1991): 58F03, 58F19, 58F34, 30F35


Additional Information

Yunping Jiang
Affiliation: Department of Mathematics, Queens College of CUNY, Flushing, New York 11367
Email: yungc@yunping.math.qc.edu, yungc@qcunix.acc.qc.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01487-0
Received by editor(s): April 28, 1992
Received by editor(s) in revised form: March 6, 1995
Additional Notes: The author is partially supported by PSC-CUNY awards (6-64053 and 6-65348) and an NSF grant (DMS-9400974).
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society