Double minimality, entropy and disjointness with all minimal systems

Piotr Oprocha

doi:10.3934/dcds.2019011

Article Contents

2019, Volume 39, Issue 1: 263-275. Doi: 10.3934/dcds.2019011

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Double minimality, entropy and disjointness with all minimal systems

Piotr Oprocha^1,2,

1.
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
2.
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

Received: November 30, 2017

Revised: June 30, 2018

Published: December 2018

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Abstract

In this paper we propose a new sufficient condition for disjointness with all minimal systems.

Using proposed approach we construct a transitive dynamical system $(X,T)$ disjoint with every minimal system and such that the set of transfer times $N(x,U)$ is not an $\text{IP}^*$-set for some nonempty open set $U\subset X$ and every $x∈ X$. This example shows that the new condition sharpens sufficient conditions for disjointness below previous bounds. In particular our approach does not rely on distality of points or sets.

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Mathematics Subject Classification: Primary: 37B20; Secondary: 37B05, 37B40.

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