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Transactions of the American Mathematical Society

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There are no piecewise linear maps of type $2^{\infty }$

Authors: Víctor Jiménez López and L'ubomír Snoha
Journal: Trans. Amer. Math. Soc. 349 (1997), 1377-1387
MSC (1991): Primary 58F08; Secondary 26A18, 54H20
MathSciNet review: 1389785
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Abstract: The aim of this paper is to show that there are no piecewise linear maps of type $2^{\infty }$. For this purpose we use the fact that any piecewise monotone map of type $2^{\infty }$ has an infinite $\omega $-limit set which is a subset of a doubling period solenoid. Then we prove that piecewise linear maps cannot have any doubling period solenoids.

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Additional Information

Víctor Jiménez López
Affiliation: Departamento de Matem\aaa ticas, Universidad de Murcia, Campus de Espinardo, Aptdo. de Correos 4021, 30100 Murcia, Spain

L'ubomír Snoha
Affiliation: Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Bansk\aaa Bystrica, Slovakia

Keywords: Asymptotically periodic point, piecewise linear map, piecewise monotone map, solenoid, wandering interval, $\omega $-limit set
Received by editor(s): October 10, 1994
Additional Notes: A part of the work on this paper was done during the stay of the second author at the University of Murcia. The invitation and the support of this institution is gratefully acknowledged.
This work has been partially supported by the DGICYT grant numbers PB91-0575 and PB94-1159 and by the Slovak grant agency, grant number 1/1470/1994.
The main result of this paper was announced at the “Thirty years after Sharkovskii’s Theorem. New perspectives" Conference, held in La Manga (Murcia), Spain, June 13-17th, 1994.
The authors are greatly indebted to the referee for many helpful suggestions which enabled them to shorten and simplify the paper.
Article copyright: © Copyright 1997 American Mathematical Society