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A mixing completely scrambled system exists

Published online by Cambridge University Press:  04 May 2017

JAN P. BOROŃSKI ,
JIŘÍ KUPKA  and
PIOTR OPROCHA
JAN P. BOROŃSKI
Affiliation:
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland email jan.boronski@osu.cz, oprocha@agh.edu.pl National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic email jiri.kupka@osu.cz
JIŘÍ KUPKA
Affiliation:
National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic email jiri.kupka@osu.cz
PIOTR OPROCHA
Affiliation:
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland email jan.boronski@osu.cz, oprocha@agh.edu.pl National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic email jiri.kupka@osu.cz
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Abstract

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that, for any integer $n\geq 1$, there is a continuum of topological dimension $n$ supporting a transitive completely scrambled homeomorphism and an $n$-dimensional compactum supporting a weakly mixing completely scrambled homeomorphism. This solves a 15-year-old open problem.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 1 , January 2019 , pp. 62 - 73
Copyright
© Cambridge University Press, 2017 

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