Minimal Non-invertible Maps on the Pseudo-Circle
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Minimal Non-invertible Maps on the Pseudo-Circle Jan P. Boronski ´ 1,2 · Judy Kennedy3 · Xiao-Chuan Liu4 · Piotr Oprocha1,2
Received: 7 April 2020 / Revised: 5 July 2020 © The Author(s) 2020
Abstract In this article we show that R.H. Bing’s pseudo-circle admits a minimal non-invertible map. This resolves a conjecture raised by Bruin, Kolyada and Snoha in the negative. The main tool is a variant of the Denjoy–Rees technique, further developed by Béguin–Crovisier–Le Roux, combined with detailed study of the structure of the pseudo-circle. This is the first example of a planar 1-dimensional space that admits both minimal homeomorphisms and minimal noninvertible maps. Keywords Pseudo-circle · Pseudoarc · Minimal · Noninvertible Mathematics Subject Classification 37B05 · 37B20 · 37B45
1 Introduction In the late 1960s J. Auslander raised a question concerning the existence of minimal noninvertible maps. Since then examples of such maps have been given, but the question as for what spaces such maps exist remains particularly interesting for compact and connected spaces
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Piotr Oprocha [email protected] Jan P. Boro´nski [email protected] Judy Kennedy [email protected] Xiao-Chuan Liu [email protected]
1
Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
2
National Supercomputing Centre IT4Innovations, Institute for Research and Applications of Fuzzy Modelling, Division of the University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic
3
Department of Mathematics, Lamar University, PO Box 10047, Beaumont, TX 77710, USA
4
Instituto de Matemática e Estatística da Universidade de São Paulo, R. do Matão, 1010 - Vila Universitaria, São Paulo, Brasil
123
Journal of Dynamics and Differential Equations
(continua). In the present paper we are going to give the first example of a 1-dimensional space in the plane that admits both minimal homeomorphisms and minimal noninvertible maps. In general, for compact spaces the existence of an invertible minimal map is independent of the existence of a noninvertible one, and there exist spaces that admit both types of maps, either one of them, or none [26]. In [1] Auslander and Yorke observed that the Cantor set admits minimal maps of both kinds. In Kolyada et al. [26] showed how to obtain minimal noninvertible maps on the 2-torus from a class of minimal homeomorphisms. An analogous result was obtained in [39] by Sotola and Trofimchuk for the Klein bottle. In [9] Clark, the first and the last mentioned authors showed the existence of both types of minimal maps for any compact finite-dimensional metric space that admits an aperiodic minimal flow. The first author showed in [10] the existence of minimal noninvertible maps for all compact manifolds of dimension at least 2, that admit minimal homeomorphisms. Auslander and Katznelson showed that the circle admits no minimal noninvertible maps, despite supporting minimal homeomorphisms [3]. In this context Bruin, Kolyada and Snoha asked the following questi
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