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A non-Borel special alpha-limit set in the square

Part of: Set theory Topological dynamics Low-dimensional dynamical systems

Published online by Cambridge University Press:  22 July 2021

STEPHEN JACKSON ,
BILL MANCE  and
SAMUEL ROTH Open the ORCID record for SAMUEL ROTH [Opens in a new window]
STEPHEN JACKSON
Affiliation:
Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle, #311430, Denton, Texas76203-5017, USA (e-mail: [email protected])
BILL MANCE
Affiliation:
Uniwersytet im. Adama Mickiewicza w Poznaniu, Collegium Mathematicum, ul. Umultowska 87, 61-614, Poznań Poland (e-mail: [email protected])
SAMUEL ROTH*
Affiliation:
Mathematical Institute of the Silesian University in Opava, Na Rybničku 1, 74601, Opava, Czech Republic
*
e-mail: [email protected]
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Abstract

We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

Keywords

special alpha limit settriangular map of the squarenon-Borel analytic set

MSC classification

Primary: 03E15: Descriptive set theory 37E99: None of the above, but in this section
Secondary: 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2550 - 2560
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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