# American Institute of Mathematical Sciences

January  2019, 39(1): 263-275. doi: 10.3934/dcds.2019011

## Double minimality, entropy and disjointness with all minimal systems

 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  December 2017 Revised  July 2018 Published  October 2018

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.

Citation: Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011
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