zbMATH — the first resource for mathematics

Dynamical stability of the typical continuous function. (English) Zbl 1150.26002
Studying the iterative stability of \(f\in C(I,I)\) we can consider the union of all \(\omega \) limit sets \(\Lambda (f)=\bigcup \omega (x,f)\) as well as the system of all \(\omega \) limit sets \(\Omega (f)=\{\omega (x,f)\:x\in I \}\). If we take corresponding maps \(\Lambda \: C(I,I)\to (K,H)\) and \(\Omega \: C(I,I) \to (K^*,H^*)\) where \((K,H)\) is the class of nonempty closed sets \(K\) in \(I\) endowed with Hausdorff metric \(H\) and \((K^*,H^*)\) is the system of nonempty closed subsets of \(K\) there are several results characterizing points of continuity of \(\Lambda \) and \(\Omega \). Present paper shows that both \(\Lambda \) and \(\Omega \) are continuous on a residual subset of \(C(I,I)\). Further it is shown that the chaotic nature of \(f\) does not depend in general on the continuity of these maps.

26A18 Iteration of real functions in one variable
Full Text: EuDML
[1] BLOCK L.-COPPEL W.: Dynamics in One Dimension. Lecture Notes in Math. 1513, Springer-Verlag, New York, 1991.
[2] BLOKH A.-BRUCKNER A. M.-HUMKE P. D.-SMÍTAL J.: The space of \(\omega\)-limit sets of a continuous map of the interval. Trans. Amer. Math. Soc. 348 (1996), 1357-1372. · Zbl 0860.54036
[3] BRUCKNER A. M.: Stability in the family of \(w\)-limit sets of continuous self maps of the interval. Real Anal. Exchange 22 (1997), 52-57. · Zbl 0888.26003
[4] BRUCKNER A. M.-BRUCKNER J. B.-THOMSON B. S.: Real Analysis. Prentice-Hall International, Upper Saddle River, NJ, 1997. · Zbl 0872.26001
[5] BRUCKNER A. M.-CEDER J. G.: Chaos in terms of the map \(x \mapsto\omega(x,f). Pacific J. Math. 156 (1992), 63-96.\) · Zbl 0728.58020
[6] BRUCKNER A. M.-SMITAL J.: A characterization of \(\omega\)-limit sets of maps of the interval with zero topological entropy. Ergodic Theory Dynam. Systems 13 (1993), 7-19. · Zbl 0788.58021
[7] FEDORENKO V.-SARKOVSKII A.-SMITAL J.: Characterizations of weakly chaotic maps of the interval. Proc. Amer. Math. Soc. 110 (1990), 141-148. · Zbl 0728.26008
[8] LI T.-YORKE J.: Period three implies chaos. Amer. Math. Monthly 82 (1975), 985-992. · Zbl 0351.92021
[9] SMITAL J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc. 297 (1986), 269-282. · Zbl 0639.54029
[10] SMITAL J.-STEELE T. H.: Stability of dynamical structures under perturbation of the generating function. · Zbl 1161.37017
[11] STEELE T. H.: Iterative stability in the class of continuous functions. Real Anal. Exchange 24 (1999), 765-780. · Zbl 0967.26005
[12] STEELE T. H.: Notions of stability for one-dimensional dynamical systems. Int. Math. J. 1 (2002), 543-555. · Zbl 1221.37085
[13] STEELE T. H.: The persistence of \(\omega\)-limit sets under perturbation of the generating function. Real Anal. Excange 26 (2000), 421-428. · Zbl 1056.26019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.