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Archive for Mathematical Logic

, Volume 57, Issue 1–2, pp 37–71 | Cite as

On some dynamical aspects of NIP theories

  • Alireza Mofidi
Article

Abstract

We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems and also to the zero entropy property of automorphisms. These results give a correspondence between some notions of tameness in model theory and ergodic theory. Moreover, we study the concept of symbolic representation and consider it in some well known mathematical objects such as the circle group, Bohr sets, Sturmian sequences, the structure \((\mathbb {Z},+,U)\), and random graphs with a model theoretic point of view in mind. We establish certain characterizations for stability theoretic dividing lines, such as independence property, order property and strict order property in terms of associated symbolic representations. At the end, we propose some applications of symbolic representations and these characterizations by giving a proof for a classical theorem by Shelah and also introducing some invariants associated to the types and elements of models.

Keywords

NIP theories Stability theory Measure preserving actions Dynamical systems Entropy Symbolic dynamics 

Mathematics Subject Classification

03C45 37A05 37B10 37A35 03C95 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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