Randomization of Sharkovskii-type theorems

Andres, Jan

doi:10.1090/S0002-9939-07-09242-8

Randomization of Sharkovskii-type theorems
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Proc. Amer. Math. Soc. 136 (2008), 1385-1395 Request permission

Erratum: Proc. Amer. Math. Soc. 136 (2008), 3733-3734.

Abstract:

We formulate an abstract scheme for the randomization of Sharkovskii-type theorems via transformation to the deterministic case. In particular, Sharkovskii-type theorems for scalar differential equations can be randomized in this way. A random version of the standard Sharkovskii theorem is presented explicitly. Many remarks, comments and illustrating examples are supplied.

References

Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Low-dimensional combinatorial dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1687–1704. Discrete dynamical systems. MR 1728729, DOI 10.1142/S0218127499001188
Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Combinatorial dynamics and entropy in dimension one, 2nd ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1807264, DOI 10.1142/4205
J. Andres, T. Fürst, and K. Pastor, Period two implies all periods for a class of ODEs: a multivalued map approach, Proc. Am. Math. Soc., 135, (2007) no. 10, 3187–3191.
—, Sharkovskii’s theorem, differential inclusions, and beyond, Submitted.
Jan Andres and Lech Górniewicz, Topological fixed point principles for boundary value problems, Topological Fixed Point Theory and Its Applications, vol. 1, Kluwer Academic Publishers, Dordrecht, 2003. MR 1998968, DOI 10.1007/978-94-017-0407-6
Jan Andres and Karel Pastor, A version of Sharkovskii’s theorem for differential equations, Proc. Amer. Math. Soc. 133 (2005), no. 2, 449–453. MR 2093067, DOI 10.1090/S0002-9939-04-07627-0
J. Andres, K. Pastor, and P. Šnyrychová, A multivalued version of Sharkovskii’s theorem holds with at most two exceptions, J. Fixed Point Th. Appl., 2 (2007) 153–170.
Jan Andres, Pavla Šnyrychová, and Piotr Szuca, Sharkovskii’s theorem for connectivity $G_\delta$-relations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 8, 2377–2393. MR 2266021, DOI 10.1142/S0218127406016136
M. Carme Leseduarte and Jaume Llibre, On the full periodicity kernel for one-dimensional maps, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 101–126. MR 1676930, DOI 10.1017/S0143385799120984
Roberta Fabbri, Tobias Jäger, Russel Johnson, and Gerhard Keller, A Sharkovskii-type theorem for minimally forced interval maps, Topol. Methods Nonlinear Anal. 26 (2005), no. 1, 163–188. MR 2179355, DOI 10.12775/TMNA.2005.029
Lech Górniewicz, Topological fixed point theory of multivalued mappings, 2nd ed., Topological Fixed Point Theory and Its Applications, vol. 4, Springer, Dordrecht, 2006. MR 2238622
Shouchuan Hu and Nikolas S. Papageorgiou, Handbook of multivalued analysis. Vol. I, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, 1997. Theory. MR 1485775, DOI 10.1007/978-1-4615-6359-4
Peter E. Kloeden, On Sharkovsky’s cycle coexistence ordering, Bull. Austral. Math. Soc. 20 (1979), no. 2, 171–177. MR 557223, DOI 10.1017/S0004972700010819
Marc Klünger, Periodicity and Sharkovsky’s theorem for random dynamical systems, Stoch. Dyn. 1 (2001), no. 3, 299–338. MR 1859009, DOI 10.1142/S0219493701000199
Helga Schirmer, A topologist’s view of Sharkovsky’s theorem, Houston J. Math. 11 (1985), no. 3, 385–395. MR 808654
A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Proceedings of the Conference “Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), 1995, pp. 1263–1273. Translated from the Russian [Ukrain. Mat. Zh. 16 (1964), no. 1, 61–71; MR0159905 (28 #3121)] by J. Tolosa. MR 1361914, DOI 10.1142/S0218127495000934
Piotr Szuca, Sharkovskiĭ’s theorem holds for some discontinuous functions, Fund. Math. 179 (2003), no. 1, 27–41. MR 2028925, DOI 10.4064/fm179-1-3