Osaka Journal of Mathematics

Periodicity of a sequence of local fixed point indices of iterations

Grzegorz Graff and Piotr Nowak-Przygodzki

Full-text: Open access


The classical theorem of Shub and Sullivan states that a sequence of local fixed point indices of iterations of a $C^1$ self-map of $\mathbb{R}^m$ is periodic. The paper generalizes this result to a wider class of maps.

Article information

Osaka J. Math., Volume 43, Number 3 (2006), 485-495.

First available in Project Euclid: 25 September 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C25: Fixed points, periodic points, fixed-point index theory
Secondary: 37B30: Index theory, Morse-Conley indices


Graff, Grzegorz; Nowak-Przygodzki, Piotr. Periodicity of a sequence of local fixed point indices of iterations. Osaka J. Math. 43 (2006), no. 3, 485--495.

Export citation


  • I.K. Babenko and S.A. Bogatyi: The behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv. 38 (1992), 1--26.
  • S.N. Chow, J. Mallet-Paret and J.A. Yorke: A periodic orbit index which is a bifurcation invariant; in Geometric Dynamics (Rio de Janeiro, 1981), Springer Lect. Notes in Math. 1007, Berlin, 1983, 109--131.
  • A. Dold: Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419--435.
  • G. Graff: Indices of iterations and periodic points of simplicial maps of smooth type, Topology Appl. 117 (2002), 77--87.
  • G. Graff and P. Nowak-Przygodzki: Fixed point indices of iterations of planar homeo-morphisms, Topol. Methods Nonlinear Anal. 22 (2003), 159--166.
  • J. Jezierski and W. Marzantowicz: Homotopy Methods in Topological Fixed and Periodic Points Theory, Kluwer Academic Pub., Dordrecht, 2005.
  • A. Katok and A. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, Cambridge, 1995.
  • W. Marzantowicz: Finding periodic points of a smooth mapping using Lefschetz numbers of its iterations, Russ. Math. 41 (1997), 80--89.
  • W. Marzantowicz and P. Przygodzki: Finding periodic points of a map by use of a $k$-adic expansion, Discrete Contin. Dynam. Systems 5 (1999), 495--514.
  • T. Matsuoka: The number of periodic points of smooth maps, Ergod. Th. Dynam. Sys. 9 (1989), 153--163.
  • T. Matsuoka and H. Shiraki: Smooth maps with finitely many periodic points, Mem. Fac. Sci., Kochi Univ. (Math) 11 (1990), 1--6.
  • F.R. Ruiz del Portal and J.M. Salazar: Fixed point index of iterations of local homeomorphisms of the plane: a Conley index approach, Topology 41 (2002), 1199--1212.
  • M. Shub and P. Sullivan: A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189--191.
  • H-W. Siegberg: A remark on the theorem of Shub and Sullivan, Topology 23 (1984), 157--160.