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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Chaotic difference equations: generic aspects
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by Hans Willi Siegberg PDF
Trans. Amer. Math. Soc. 279 (1983), 205-213 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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