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$C^k$ 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
MathSciNet review: 1372046
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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.

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  • [ACL] Valentine Afraimovich, Shui-Nee Chow, and Weishi Liu, Lorenz-type attractors from codimension one bifurcation, J. Dynam. Differential Equations 7 (1995), no. 2, 375–407. MR 1336467, 10.1007/BF02219362
  • [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. Belitskiĭ, Smooth classification of one-dimensional diffeomorphisms with hyperbolic fixed points, Sibirsk. Mat. Zh. 27 (1986), no. 6, 21–24 (Russian). MR 883578
  • [IY] Yu. S. Il′yashenko and S. Yu. Yakovenko, Nonlinear Stokes phenomena in smooth classification problems, Nonlinear Stokes phenomena, Adv. Soviet Math., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 235–287. MR 1206045
  • [KH] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • [KCG] Marek Kuczma, Bogdan Choczewski, and Roman Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990. MR 1067720
  • [MS] 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
  • [NPT] S. Newhouse, J. Palis, and F. Takens. Bifurcations and Stability of Families of Diffeomorphisms. Bulletin of the AMS, 82 (1976) 499.

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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

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