Characterizations of turbulent one-dimensional mappings via 𝜔-limit sets

Evans, Michael J.; Humke, Paul D.; Lee, Cheng Ming; O’Malley, Richard J.

doi:10.1090/S0002-9947-1991-1010884-3

Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets
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by Michael J. Evans, Paul D. Humke, Cheng Ming Lee and Richard J. O’Malley PDF

Trans. Amer. Math. Soc. 326 (1991), 261-280 Request permission

Corrigendum: Trans. Amer. Math. Soc. 333 (1992), 939-940.

Abstract:

The structure of $\omega$-limit sets for nonturbulent functions is studied, and various characterizations for turbulent and chaotic functions are obtained. In particular, it is proved that a continuous function mapping a compact interval into itself is turbulent if and only if there exists an $\omega$-limit set which is a unilaterally convergent sequence

References

S. J. Agronsky, A. M. Bruckner, J. G. Ceder, and T. L. Pearson, The structure of $\omega$-limit sets for continuous functions, Real Anal. Exchange 15 (1989/90), no. 2, 483–510. MR 1059418, DOI 10.2307/44152033
L. S. Block and W. A. Coppel, Stratification of continuous maps of an interval, Trans. Amer. Math. Soc. 297 (1986), no. 2, 587–604. MR 854086, DOI 10.1090/S0002-9947-1986-0854086-8

Continuous maps of an interval

Robert L. Devaney, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR 811850
K. S. Sibirsky, Introduction to topological dynamics, Noordhoff International Publishing, Leiden, 1975. Translated from the Russian by Leo F. Boron. MR 0357987
A. N. Šarkovskiĭ, The behavior of the transformation in the neighborhood of an attracting set, Ukrain. Mat. Ž. 18 (1966), no. 2, 60–83 (Russian). MR 0212784

The partially ordered system of attracting sets

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