Characterizations of weakly chaotic maps of the interval
Authors:
V. V. Fedorenko, A. N. Šarkovskii and J. Smítal
Journal:
Proc. Amer. Math. Soc. 110 (1990), 141148
MSC:
Primary 58F20; Secondary 28D20, 54H20, 58F08, 58F13
MathSciNet review:
1017846
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Abstract: We prove, among others, the following relations between notions of chaos for continuous maps of the interval: (i) A map is not chaotic in the sense of Li and Yorke iff restricted to the set of its limit points is stable in the sense of Ljapunov. (ii) The topological entropy of is zero iff restricted to the set of chain recurrent points is not chaotic in the sense of Li and Yorke, and this is iff every trajectory is approximable by trajectories of periodic intervals.
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A. M. Blokh, On limit behavior of onedimensional dynamical systems, Uspekhi Mat. Nauk 37 (1982), 137138. (Russian) MR 643772 (83i:58082)
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L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391398. MR 539925 (80m:58031)
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R. Bowen, Topological entropy and Axiom A, Global Analysis, Proc. Sympos. Pure Math. 14 (1970), 2341. MR 0262459 (41:7066)
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K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283292. MR 854575 (87k:58178)
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M. Kuchta and J. Smítal, Two point scrambled set implies chaos, Proc. Europ. Conf. on Iteration Theory (Caldas de Malavella, Spain, 1987), World Scientific, 1989, 427430. MR 1085314 (91j:58112)
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M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory Dynamical Systems 8 (1988), 421424. MR 961740 (90a:58118)
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D. Preiss and J. Smítal, A characterization of nonchaotic maps of the interval stable under small perturbations, Trans. Amer. Math. Soc. (to appear). MR 997677 (90f:58100)
 [8]
A. N. Šarkovskii, Nonwandering points and the center of a continuous mapping of the line into itself, Dopovidi Akad. Nauk Ukraiin RSR Ser. A. No. 7 (1964), 865868. (Ukrainian) MR 0165178 (29:2467)
 [9]
, The partially ordered system of attracting sets, Soviet Math. Dokl.7 (1966), 13841386.
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A. N. Šarkovskii, J. L. Majstrenko, and B. J. Romanenko, Difference equations and their applications, Naukova Dumka, Kiev, 1986. (Russian) MR 895825 (88k:39007)
 [11]
A. N. Šarkovskii, S. F. Koljada, A. G. Sivak, and V. V. Fedorenko, Dynamics of onedimensional mappings, Naukova Dumka, Kiev, 1989. (Russian) MR 1036027 (91k:58065)
 [12]
J. Smítal, Chaotic maps with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269282.
 [13]
J. Xiong, Set of almost periodic points of a continuous selfmap of an interval, Acta. Math. Sinica (N.S.) 2 (1986), 7377. MR 877371 (88d:58093)
 [14]
M. B. Verejkina and A. N. Šarkovskii, Recurrence in onedimensional dynamical systems, Approx. and Qualitative Methods of the Theory of DifferentialFunctional Equations, Inst. Math. Akad. Nauk USSR, Kiev, 1983, pp. 3546. (Russian) MR 753681 (85m:58149)
 [1]
 A. M. Blokh, On limit behavior of onedimensional dynamical systems, Uspekhi Mat. Nauk 37 (1982), 137138. (Russian) MR 643772 (83i:58082)
 [2]
 L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391398. MR 539925 (80m:58031)
 [3]
 R. Bowen, Topological entropy and Axiom A, Global Analysis, Proc. Sympos. Pure Math. 14 (1970), 2341. MR 0262459 (41:7066)
 [4]
 K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283292. MR 854575 (87k:58178)
 [5]
 M. Kuchta and J. Smítal, Two point scrambled set implies chaos, Proc. Europ. Conf. on Iteration Theory (Caldas de Malavella, Spain, 1987), World Scientific, 1989, 427430. MR 1085314 (91j:58112)
 [6]
 M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory Dynamical Systems 8 (1988), 421424. MR 961740 (90a:58118)
 [7]
 D. Preiss and J. Smítal, A characterization of nonchaotic maps of the interval stable under small perturbations, Trans. Amer. Math. Soc. (to appear). MR 997677 (90f:58100)
 [8]
 A. N. Šarkovskii, Nonwandering points and the center of a continuous mapping of the line into itself, Dopovidi Akad. Nauk Ukraiin RSR Ser. A. No. 7 (1964), 865868. (Ukrainian) MR 0165178 (29:2467)
 [9]
 , The partially ordered system of attracting sets, Soviet Math. Dokl.7 (1966), 13841386.
 [10]
 A. N. Šarkovskii, J. L. Majstrenko, and B. J. Romanenko, Difference equations and their applications, Naukova Dumka, Kiev, 1986. (Russian) MR 895825 (88k:39007)
 [11]
 A. N. Šarkovskii, S. F. Koljada, A. G. Sivak, and V. V. Fedorenko, Dynamics of onedimensional mappings, Naukova Dumka, Kiev, 1989. (Russian) MR 1036027 (91k:58065)
 [12]
 J. Smítal, Chaotic maps with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269282.
 [13]
 J. Xiong, Set of almost periodic points of a continuous selfmap of an interval, Acta. Math. Sinica (N.S.) 2 (1986), 7377. MR 877371 (88d:58093)
 [14]
 M. B. Verejkina and A. N. Šarkovskii, Recurrence in onedimensional dynamical systems, Approx. and Qualitative Methods of the Theory of DifferentialFunctional Equations, Inst. Math. Akad. Nauk USSR, Kiev, 1983, pp. 3546. (Russian) MR 753681 (85m:58149)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010178465
PII:
S 00029939(1990)10178465
Article copyright:
© Copyright 1990
American Mathematical Society
