Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$C^k$ conjugacy of 1-d diffeomorphisms
with periodic points


Author: Todd R. Young
Journal: Proc. Amer. Math. Soc. 125 (1997), 1987-1995
MSC (1991): Primary 34C35, 58C25
DOI: https://doi.org/10.1090/S0002-9939-97-03783-0
MathSciNet review: 1372046
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the set of heteroclinic orbits between two periodic orbits of saddle-node type induces functional moduli which are completely contained in a new `transition map'. For one-dimensional $C^2$ diffeomorphisms with saddle-node periodic points, two such diffeomorphisms are $C^2$ conjugated if and only if the transition maps of their heteroclinic orbits are the same. An equivalent transition map is given for $C^k$ diffeomorphisms with hyperbolic periodic points, and it is shown that this transition map is an invariant of $C^k$ conjugation. However, in this case the transition map alone is sufficient to guarantee conjugacy only in a limited sense.


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

  • [ACL] V.S. Afraimovich, S.-N. Chow and W.-S. Liu, Lorenz type attractors from a codimensional one bifurcation, Diff. Eqns. Dynam. Sys., 7 (1995), 375-407. MR 96c:58097
  • [ALY] V.S. Afraimovich, W.-S. Liu, and T. Young, Conventional multipliers for homoclinic orbits, Nonlinearity, 9 (1996), 115-136. CMP 96:08
  • [Be] G.R. Belitski, Smooth classification of one-dimensional diffeomorphisms with hyperbolic fixed points, Sibirsk. Mat. Zh., 27 (1986), 21-24 [Russian]. Trans. in Siberian Mat. J., 27 (1986), 801-804. MR 88f:58117
  • [IY] Y. Il'yashenko and S. Yakovenko, Nonlinear Stokes phenomena in smooth classification problems, Advances in Soviet Mathematics, 14 (1993), 235-287. MR 94f:58099
  • [KH] A. Katok and B. Hasselblat, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995. MR 96c:58055
  • [KCG] M. Kuozma, B. Choczewski, and R. Ger, Iterative Functional Equations, Cambridge University Press, 1990. MR 92f:39002
  • [MS] W. de Melo and S. van Strien, One-dimensional Dynamics Springer-Verlag: New York, 1993. MR 95a:58035
  • [NPT] S. Newhouse, J. Palis, and F. Takens. Bifurcations and Stability of Families of Diffeomorphisms. Bulletin of the AMS, 82 (1976) 499.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34C35, 58C25

Retrieve articles in all journals with MSC (1991): 34C35, 58C25


Additional Information

Todd R. Young
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Address at time of publication: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730
Email: young@math.nwu.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03783-0
Keywords: Smooth conjugacy, global conjugacy
Received by editor(s): June 14, 1995
Received by editor(s) in revised form: January 9, 1996
Additional Notes: The author was partially supported by AFOSR grant #F49620-93-1-0147.
Communicated by: Mary Rees
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society