There are no piecewise linear maps of type

Authors:
Víctor Jiménez López and L'ubomír Snoha

Journal:
Trans. Amer. Math. Soc. **349** (1997), 1377-1387

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

DOI:
https://doi.org/10.1090/S0002-9947-97-01801-1

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 . For this purpose we use the fact that any piecewise monotone map of type has an infinite -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

Email:
vjimenez@fcu.um.es

**L'ubomír Snoha**

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

Email:
snoha@bb.sanet.sk

DOI:
https://doi.org/10.1090/S0002-9947-97-01801-1

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