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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Chaotic difference equations: generic aspects


Author: Hans Willi Siegberg
Journal: Trans. Amer. Math. Soc. 279 (1983), 205-213
MSC: Primary 58F14; Secondary 58F08, 58F12
DOI: https://doi.org/10.1090/S0002-9947-1983-0704610-8
MathSciNet review: 704610
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Abstract: It is shown that in the set of all continuous selfmaps of a compact acyclic polyhedron (i.e. the homology groups of the space vanish in all dimensions $ > 0$) the chaotic maps form a dense set. The notion of chaos used here is that of Li and Yorke. If this notion is slightly weakened ("almost chaotic") the density result can be improved by the theorem that the set of almost chaotic (continuous) selfmaps of a compact acyclic polyhedron $ P$ contains a residual subset of the space of all continuous selfmaps of $ P$. Moreover, the topological entropy of such a generic almost chaotic map is shown to be infinite. The basic ingredients of the proofs are from fixed point index theory.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0704610-8
Article copyright: © Copyright 1983 American Mathematical Society