Minimal Non-invertible Maps on the Pseudo-Circle

Author(s)

Boronski, Jan P. ^{[1, 2]} ; Kennedy, Judy ^[3] ; Liu, Xiao-Chuan ^[4] ; Oprocha, Piotr ^{[1, 2]}

Affiliation

^[1] AGH Univ Sci & Technol, Fac Appl Math, Al Mickiewicza 30, PL-30059 Krakow - Poland

^[2] Div Univ Ostrava, Natl Supercomp Ctr IT4Innovat, Inst Res & Applicat Fuzzy Modelling, 30 Dubna 22, Ostrava 70103 - Czech Republic

^[3] Lamar Univ, Dept Math, POB 10047, Beaumont, TX 77710 - USA

^[4] Univ Sao Paulo, Inst Matemat & Estat, R Matao 1010 Vila Univ, Sao Paulo - Brazil

• Document type: Journal article
• Source: Journal of Dynamics and Differential Equations; v. 33, n. 4, p. 1897-1916, DEC 2021.
• Web of Science Citations: 1

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 Beguin-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. (AU)

• FAPESP's process: 18/03762-2 - Topological dynamical system on surfaces
• Grantee: Xiaochuan Liu
• Support Opportunities: Scholarships in Brazil - Post-Doctoral