All maps of type are boundary maps

Authors:
Víctor Jiménez López and L'ubomír Snoha

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1667-1673

MSC (1991):
Primary 26A18; Secondary 58F08, 54H20

MathSciNet review:
1342034

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a continuous map of an interval into itself having periodic points of period for all and no other periods. It is shown that every neighborhood of contains a map such that the set of periods of the periodic points of is finite. This answers a question posed by L. S. Block and W. A. Coppel.

**[AKM]**R. L. Adler, A. G. Konheim, and M. H. McAndrew,*Topological entropy*, Trans. Amer. Math. Soc.**114**(1965), 309–319. MR**0175106**, 10.1090/S0002-9947-1965-0175106-9**[ALM]**Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz,*Combinatorial dynamics and entropy in dimension one*, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR**1255515****[Bl]**Louis Block,*Stability of periodic orbits in the theorem of Šarkovskii*, Proc. Amer. Math. Soc.**81**(1981), no. 2, 333–336. MR**593484**, 10.1090/S0002-9939-1981-0593484-8**[BC]**L. S. Block and W. A. Coppel,*Dynamics in one dimension*, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR**1176513****[BF]**Rufus Bowen and John Franks,*The periodic points of maps of the disk and the interval*, Topology**15**(1976), no. 4, 337–342. MR**0431282****[Co]**W. A. Coppel,*The solution of equations by iteration*, Proc. Cambridge Philos. Soc.**51**(1955), 41–43. MR**0066444****[FS]**V. V. Fedorenko and J. Smítal,*Maps of the interval Ljapunov stable on the set of nonwandering points*, Acta Math. Univ. Comenian. (N.S.)**60**(1991), no. 1, 11–14. MR**1120591****[Ge]**Tomáš Gedeon,*Stable and nonstable nonchaotic maps of the interval*, Math. Slovaca**41**(1991), no. 4, 379–391. MR**1149045****[JS]**V. Jiménez López and L'. Snoha,*There are no piecewise linear maps of type*, Trans. Amer. Math. Soc. (to appear). CMP**96:12****[Kl]**Peter E. Kloeden,*Chaotic difference equations are dense*, Bull. Austral. Math. Soc.**15**(1976), no. 3, 371–379. MR**0432829****[Mi]**Michał Misiurewicz,*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****[Pr]**Chris Preston,*Iterates of piecewise monotone mappings on an interval*, Lecture Notes in Mathematics, vol. 1347, Springer-Verlag, Berlin, 1988. MR**969131****[Sh1]**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**, 10.1142/S0218127495000934**[Sh2]**A. N. Sharkovskii,*On cycles and the structure of a continuous mapping*, Ukrain. Math. Zh.**17**(1965), 104-111 (Russian).**[Sm]**J. Smítal,*Chaotic functions with zero topological entropy*, Trans. Amer. Math. Soc.**297**(1986), no. 1, 269–282. MR**849479**, 10.1090/S0002-9947-1986-0849479-9**[W]**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
26A18,
58F08,
54H20

Retrieve articles in all journals with MSC (1991): 26A18, 58F08, 54H20

Additional Information

**Víctor Jiménez López**

Affiliation:
Departamento de Matemá ticas, Universidad de Murcia, Campus de Espinardo, Aptdo. de Correos 4021, 30100 Murcia, Spain

Email:
vjimenez@fcu.um.es

**L'ubomír Snoha**

Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovsk\ee ho 40, 974 01 Bansk\aaa Bystrica, Slovakia

Email:
snoha@bb.sanet.sk

DOI:
http://dx.doi.org/10.1090/S0002-9939-97-03452-7

Keywords:
Map of type $2^{\infty }$,
periodic point,
solenoid

Received by editor(s):
February 27, 1995

Received by editor(s) in revised form:
May 2, 1995

Additional Notes:
Most of the work on this paper was done during the stay of the first author at the Matej Bel University. The invitation and the support of this institution is gratefully acknowledged. The first author has been partially supported by the DGICYT PB91-0575 and the second author by the Slovak grant agency, grant number 1/1470/94.

Communicated by:
Mary Rees

Article copyright:
© Copyright 1997
American Mathematical Society