Chaotic difference equations: generic aspects

Siegberg, Hans Willi

doi:10.1090/S0002-9947-1983-0704610-8

Chaotic difference equations: generic aspects
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by Hans Willi Siegberg

Trans. Amer. Math. Soc. 279 (1983), 205-213

DOI: https://doi.org/10.1090/S0002-9947-1983-0704610-8

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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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