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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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There are no piecewise linear maps of type $2^{\infty }$
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by Víctor Jiménez López and L’ubomír Snoha PDF
Trans. Amer. Math. Soc. 349 (1997), 1377-1387 Request permission

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 Matemáticas, 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 Banská Bystrica, Slovakia
  • MR Author ID: 250583
  • Email: snoha@bb.sanet.sk
  • 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.
  • © Copyright 1997 American Mathematical Society
  • 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