Dynamics in dimension zero A survey

Tomasz Downarowicz; Olena Karpel

doi:10.3934/dcds.2018044

Article Contents

2018, Volume 38, Issue 3: 1033-1062. Doi: 10.3934/dcds.2018044

This issue Previous Article What is topological about topological dynamics? Next Article The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations

Dynamics in dimension zero A survey

Tomasz Downarowicz^1, and
Olena Karpel^2,

1.
Faculty of Mathematics and Faculty of Fundamental Problems of Technology, Wroclaw University of Technology, Wroclaw, Poland
2.
Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine, Current address: Department of Dynamical Systems, Institute of Mathematics of Polish Academy of Sciences, Wroclaw, Poland

Received: October 31, 2016

Revised: September 30, 2017

Published: June 2018

The second author is supported by the NCN (National Science Center, Poland) Grant 2013/08/A/ST1/00275

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The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can switch from one representation to another. We also briefly review some more recent related notions: symbolic extensions, symbolic extensions with an embedding, and uniform generators. We devote a great deal of attention to marker techniques and we use them to prove two types of results: one concerning entropy and vertical data compression, and another, about the existence of isomorphic minimal models for aperiodic systems. We also introduce so-called decisiveness of Bratteli-Vershik systems and give for it a sufficient condition.

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Mathematics Subject Classification: Primary: 37B05, 37B10; Secondary: 37B40.

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