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)



Iterating maps on cellular complexes

Author: Stephen J. Willson
Journal: Trans. Amer. Math. Soc. 332 (1992), 225-240
MSC: Primary 58F13; Secondary 05C20
MathSciNet review: 1049619
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a finite simplicial complex and $ f:K \to K$ be a "skeletal" map. A digraph $ D$ is defined whose vertices correspond to the simplexes of $ K$ and whose arcs give information about the behavior of $ f$ on the simplexes. For every walk in $ D$ there exists a point of $ K$ whose iterates under $ f$ mimic the walk. Periodic walks are mimicked by a periodic point. Digraphs with uncountably many infinite walks are characterized; the corresponding maps $ f$ exhibit complicated behavior.

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

  • [AS] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1972.
  • [B] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, Berlin, 1975. MR 0442989 (56:1364)
  • [CE] P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, Basel, 1983.
  • [D] R. L. Devaney, An introduction to chaotic dynamical systems, Benjamin, Menlo Park, 1986. MR 811850 (87e:58142)
  • [Ga] F. R. Gantmacher, The theory of matrices, vol. 2, Chelsea, New York, 1959.
  • [Gu] J. Guckenheimer, Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), 133-160. MR 553966 (82c:58037)
  • [MT] J. Milnor and W. Thurston, On iterated maps of the interval. I, II, preprint, Princeton Univ., 1977. MR 970571 (90a:58083)
  • [P] W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1964), 368-378. MR 0197683 (33:5846)
  • [RF] D. F. Robinson and L. R. Foulds, Digraphs: theory and techniques, Gordon and Breach, New York, 1980. MR 580665 (81k:05055)
  • [S] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [W] P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F13, 05C20

Retrieve articles in all journals with MSC: 58F13, 05C20

Additional Information

Keywords: Simplicial complex, digraph, chaotic dynamics
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society