Iterating maps on cellular complexes

Author:
Stephen J. Willson

Journal:
Trans. Amer. Math. Soc. **332** (1992), 225-240

MSC:
Primary 58F13; Secondary 05C20

DOI:
https://doi.org/10.1090/S0002-9947-1992-1049619-8

MathSciNet review:
1049619

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Abstract | References | Similar Articles | Additional Information

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

**[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)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1049619-8

Keywords:
Simplicial complex,
digraph,
chaotic dynamics

Article copyright:
© Copyright 1992
American Mathematical Society