A zero topological entropy map with recurrent points not $F_\sigma$
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- by Petra Šindelářová PDF
- Proc. Amer. Math. Soc. 131 (2003), 2089-2096 Request permission
Abstract:
We show that there is a continuous map $\chi$ of the unit interval into itself of type $2^\infty$ which has a trajectory disjoint from the set $\operatorname {Rec}(\chi )$ of recurrent points of $\chi$, but contained in the closure of $\operatorname {Rec}(\chi )$. In particular, $\operatorname {Rec}(\chi )$ is not closed. A function $\psi$ of type $2^\infty$, with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361–366]. However, there is no trajectory contained in $\overline {\operatorname {Rec} (\psi )}\setminus \operatorname {Rec}(\psi )$, since any point in $\overline { \operatorname {Rec}(\psi )}$ is eventually mapped into $\operatorname {Rec} (\psi )$. Moreover, our construction is simpler. We use $\chi$ to show that there is a continuous map of the interval of type $2^\infty$ for which the set of recurrent points is not an $F_\sigma$ set. This example disproves a conjecture of A. N. Sharkovsky et al., from 1989. We also provide another application of $\chi$.References
- Lluís Alsedà, Moira Chas, and Jaroslav Smítal, On the structure of the $\omega$-limit sets for continuous maps of the interval, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1719–1729. Discrete dynamical systems. MR 1728731, DOI 10.1142/S0218127499001206
- Louis Block, Stability of periodic orbits in the theorem of Šarkovskii, Proc. Amer. Math. Soc. 81 (1981), no. 2, 333–336. MR 593484, DOI 10.1090/S0002-9939-1981-0593484-8
- L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513, DOI 10.1007/BFb0084762
- A. M. Bruckner and J. Smítal, A characterization of $\omega$-limit sets of maps of the interval with zero topological entropy, Ergodic Theory Dynam. Systems 13 (1993), no. 1, 7–19. MR 1213076, DOI 10.1017/S0143385700007173
- Hsin Chu and Jin Cheng Xiong, A counterexample in dynamical systems of the interval, Proc. Amer. Math. Soc. 97 (1986), no. 2, 361–366. MR 835899, DOI 10.1090/S0002-9939-1986-0835899-0
- A. Denjoy, Sur les courbes définies par les équations differentielles à la surface du tore , J. Math. Pures Appl. 11 (1932), 333-375.
- V. V. Fedorenko, Classification of simple one-dimensional dynamical systems, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 5 (1991), 30 (Russian). MR 1153696
- V. V. Fedorenko, A. N. Šarkovskii, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), no. 1, 141–148. MR 1017846, DOI 10.1090/S0002-9939-1990-1017846-5
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
- MichałMisiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
- A. N. Sharkovsky, Classification of one-dimensional dynamical systems, European Conference on Iteration Theory (Caldes de Malavella, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 42–55. MR 1085276
- A. N. Sharkovskiĭ, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dinamika odnomernykh otobrazheniĭ, “Naukova Dumka”, Kiev, 1989 (Russian). MR 1036027
- A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of one-dimensional maps, Mathematics and its Applications, vol. 407, Kluwer Academic Publishers Group, Dordrecht, 1997. Translated from the 1989 Russian original by Sivak, P. Malyshev and D. Malyshev and revised by the authors. MR 1448407, DOI 10.1007/978-94-015-8897-3
- A. N. Sharkovsky, Yu. L. Maĭstrenko, and E. Yu. Romanenko, Difference equations and their applications, Mathematics and its Applications, vol. 250, Kluwer Academic Publishers Group, Dordrecht, 1993. Translated from the 1986 Russian original by D. V. Malyshev, P. V. Malyshev and Y. M. Pestryakov. MR 1244579, DOI 10.1007/978-94-011-1763-0
- P. Šindelářová, A zero topological entropy map for which periodic points are not a $G_\delta$ set, Ergod. Th. & Dynam. Sys. 22 (2002), 947–949.
Additional Information
- Petra Šindelářová
- Affiliation: Mathematical Institute, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic
- Email: Petra.Sindelarova@math.slu.cz
- Received by editor(s): January 7, 2002
- Published electronically: February 5, 2003
- Additional Notes: This research was supported, in part, by contracts 201/00/0859 from the Grant Agency of Czech Republic and CEZ:J10/98:192400002 from the Czech Ministry of Education. The support of these institutions is gratefully acknowledged
- Communicated by: Michael Handel
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 2089-2096
- MSC (2000): Primary 26A18, 37E05
- DOI: https://doi.org/10.1090/S0002-9939-03-06971-5
- MathSciNet review: 1963754
Dedicated: Dedicated to Professor Jaroslav Smítal on the occasion of his 60th birthday