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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

MSC:

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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[1] R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309 – 319. · Zbl 0127.13102
[2] H. Amann, Lectures on some fixed point theorems, Monograf. Mat., Inst. Mat. Pura Apl., Rio de Janeiro, 1974.
[3] Louis Block, Stability of periodic orbits in the theorem of Šarkovskii, Proc. Amer. Math. Soc. 81 (1981), no. 2, 333 – 336. · Zbl 0462.54029
[4] Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401 – 414. · Zbl 0212.29201
[5] Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337 – 342. · Zbl 0346.58010
[6] Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. · Zbl 0216.19601
[7] G. J. Butler and G. Pianigiani, Periodic points and chaotic functions in the unit interval, Bull. Austral. Math. Soc. 18 (1978), no. 2, 255 – 265. · Zbl 0389.54026
[8] J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353 – 367. · Zbl 0043.38105
[9] Günther Eisenack and Christian Fenske, Fixpunkttheorie, Bibliographisches Institut, Mannheim, 1978 (German). · Zbl 0369.47001
[10] Heinz Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), no. 1, 493 – 524 (German). · JFM 55.0970.02
[11] Peter E. Kloeden, Chaotic difference equations are dense, Bull. Austral. Math. Soc. 15 (1976), no. 3, 371 – 379. · Zbl 0335.39001
[12] Andrzej Lasota and James A. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 3, 233 – 238 (English, with Russian summary). · Zbl 0357.28018
[13] T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985 – 992. · Zbl 0351.92021
[14] Frederick R. Marotto, Snap-back repellers imply chaos in \?\(^{n}\), J. Math. Anal. Appl. 63 (1978), no. 1, 199 – 223. · Zbl 0381.58004
[15] O. M. Šarkovs\(^{\prime}\)kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž. 16 (1964), 61 – 71 (Russian, with English summary).
[16] Michael Shub, Stabilité globale des systèmes dynamiques, Astérisque, vol. 56, Société Mathématique de France, Paris, 1978 (French). With an English preface and summary. · Zbl 0396.58014
[17] H. W. Siegberg, Zur komplexen Dynamik iterierter Abbildungen, Dissertation, University of Bremen, 1982. · Zbl 0561.58028
[18] P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), no. 3, 237 – 248.
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