Chaotic difference equations: generic aspects. (English) Zbl 0532.58015

This paper deals with the ”chaotic” behaviour of the equation: (1) \(x_{n+1}=f(x_ n),n\in N\), where f:\(X\to X\) is a continuous map of a compact polyhedron X into itself. The notion of chaos used here is that of Li and Yorke, i.e. it does not concern a stable (attractive) ”very complex” solution of (1). What is called ”chaos” here is related to the set of the limit points of all repulsive periodic points and their antecedents of all ranks. The author’s purpose is to extend known results of the one-dimensional case to mappings of certain higher-dimensional spaces (acyclic polyhedra), by using tools different from the familiar techniques of interval mappings. The basic ingredients of the proofs are ”fixed point index” and ”degree arguments”. It is shown that in the set of all continuous selfmaps of a compact acyclic polyhedron, the ”chaotic” maps form a dense set. If the chaos notion of Yorke-Li is slightly weakened (”almost chaotic”) the density result can be improved. Then 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 infinite.
Reviewer: C.Mira


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54H20 Topological dynamics (MSC2010)
54C05 Continuous maps
54C70 Entropy in general topology
39A10 Additive difference equations
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