Journal of Modern Dynamics (JMD)

Dynamical cubes and a criteria for systems having product extensions

Pages: 365 - 405, Volume 9, 2015      doi:10.3934/jmd.2015.9.365

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Sebastián Donoso - Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (email)
Wenbo Sun - Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, IL 60208-2730, United States (email)

Abstract: For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the usual regional proximality relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have less intricate dynamics, and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions and a computation of the automorphism group of the minimal Robinson tiling.

Keywords:  Minimal systems, dynamical cubes, product systems.
Mathematics Subject Classification:  Primary: 54H20; Secondary: 37B05.

Received: March 2015;      Revised: November 2015;      Available Online: December 2015.