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)



Period doubling and the Lefschetz formula

Author: John Franks
Journal: Trans. Amer. Math. Soc. 287 (1985), 275-283
MSC: Primary 58F20; Secondary 55M20
MathSciNet review: 766219
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article gives an application of the Lefschetz fixed point theorem to prove, under certain hypotheses, the existence of a family of periodic orbits for a smooth map. The family has points of periods $ {2^k}p$ for some $ p$ and all $ k \geq 0$. There is a version of the result for a parametrized family $ f_t$ which shows that these orbits are "connected" in parametrized space under appropriate hypotheses.

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

  • [AY] K. Alligood and J. Yorke, Cascades of period-doubling bifurcations: A prerequisite for horseshoes, Bull. Amer. Math. Soc. (to appear). MR 714994 (85b:58089)
  • [B] P. Blanchard, Invariants of the $ NPT$ isotpy classes of Morse-Smale diffeomorphism of surfaces, Duke Math. J. 47 (1980), 33-46. MR 563365 (81e:58042)
  • [Br] P. Brunovsky, On one-parameter families of diffeomorphisms. I, II, Comment. Math. Univ. Carolinae 11 (1970), 559-582; 12 (1971), 765-784. MR 0279827 (43:5548)
  • [DN] R. Devaney and Z. Nitecki, Shift automorphisms in the Henon mapping, Comment. Math. Phys. 67 (1979), 137-146. MR 539548 (80f:58035)
  • [D] A Dold, Fixed point index and fixed point theorems for Euclidean neighborhood retracts, Topology 4 (1965), 1-8. MR 0193634 (33:1850)
  • [F] J. Franks, Homology and dynamical systems, CBMS Regional Conf. Ser. in Math., no. 49, Amer. Math. Soc., Providence, R.I., 1982. MR 669378 (84f:58067)
  • [Fr] D. Fried, Periodic points and twisted coefficients (to appear). MR 730272 (85h:58142)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F20, 55M20

Retrieve articles in all journals with MSC: 58F20, 55M20

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society