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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
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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.
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G. Konheim, and M.
H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 0175106
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Soc. 81 (1981), no. 2, 333–336. MR 593484
(82b:58071), http://dx.doi.org/10.1090/S0002-9939-1981-0593484-8
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S. Block and W.
A. Coppel, Dynamics in one dimension, Lecture Notes in
Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513
(93g:58091)
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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
(55 #4283)
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W.
A. Coppel, The solution of equations by iteration, Proc.
Cambridge Philos. Soc. 51 (1955), 41–43. MR 0066444
(16,577a)
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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
(92g:58063)
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Tomáš
Gedeon, Stable and nonstable nonchaotic maps of the interval,
Math. Slovaca 41 (1991), no. 4, 379–391. MR 1149045
(92k:58074)
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V. Jiménez López and L'. Snoha, There are no piecewise linear maps of type
 , Trans. Amer. Math. Soc. (to appear). CMP 96:12
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Peter
E. Kloeden, Chaotic difference equations are dense, Bull.
Austral. Math. Soc. 15 (1976), no. 3, 371–379.
MR
0432829 (55 #5809)
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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
(81b:58033)
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Chris
Preston, Iterates of piecewise monotone mappings on an
interval, Lecture Notes in Mathematics, vol. 1347,
Springer-Verlag, Berlin, 1988. MR 969131
(89m:58109)
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N. Sharkovskiĭ, Coexistence of cycles of a continuous map of
the line into itself, Proceedings of the Conference “Thirty
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1361914 (96j:58058), http://dx.doi.org/10.1142/S0218127495000934
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J.
Smítal, Chaotic functions with zero
topological entropy, Trans. Amer. Math.
Soc. 297 (1986), no. 1, 269–282. MR 849479
(87m:58107), http://dx.doi.org/10.1090/S0002-9947-1986-0849479-9
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Peter
Walters, An introduction to ergodic theory, Graduate Texts in
Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR 648108
(84e:28017)
- [AKM]
- R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 30:5291
- [ALM]
- L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publ., Singapore, 1993. MR 95j:58042
- [Bl]
- L. Block, Stability of periodic orbits in the theorem of Sarkovskii, Proc. Amer. Math. Soc. 81 (1981), 333-336. MR 82b:58071
- [BC]
- L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math., vol. 1513, Springer, Berlin, 1992. MR 93g:58091
- [BF]
- R. Bowen and J. Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), 337-342. MR 55:4283
- [Co]
- W. A. Coppel, The solutions of equations by iteration, Proc. Camb. Phil. Soc. 51 (1955), 41-43. MR 16:577a
- [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), 11-14. MR 92g:58063
- [Ge]
- T. Gedeon, Stable and non-stable non-chaotic maps of the interval, Math. Slovaca 41 (1991), 379-391. MR 92k:58074
- [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]
- P. E. Kloeden, Chaotic difference equations are dense, Bull. Austral. Math. Soc. 15 (1976), 371-379. MR 55:5809
- [Mi]
- M. Misiurewicz, Horseshoes for mappings of an interval, Bull. Acad. Pol. Sci., Sér. Sci. Math. 27 (1979), 167-169. MR 81b:58033
- [Pr]
- C. Preston, Iterates of Piecewise Monotone Mappings on an Interval, Lecture Notes in Mathematics, vol. 1347, Springer, Berlin, 1988. MR 89m:58109
- [Sh1]
- A. N. Sharkovskii, Coexistence of cycles of a continuous mapping of the line into itself, Ukrain. Math. Zh. 16 (1964), 61-71 (Russian). MR 96j:58058
- [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. Mat. Soc. 297 (1986), 269-282. MR 87m:58107
- [W]
- P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. MR 84e:28017
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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
PII:
S 0002-9939(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
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