Synchronizing Dynamical Systems: Shift Spaces and \(K\)-Theory

Building on previous work by the author and Robin Deeley, we give a thorough presentation of the techniques developed for synchronizing dynamical systems in the special case of synchronizing shift spaces. Following work of Thomsen, we give a construction of the homoclinic and heteroclinic \(C^\ast\)...

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Bibliographic Details
Published inarXiv.org
Main Author Stocker, Andrew M
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.08.2022
Subjects
Dynamical systems
Synchronism
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Summary:Building on previous work by the author and Robin Deeley, we give a thorough presentation of the techniques developed for synchronizing dynamical systems in the special case of synchronizing shift spaces. Following work of Thomsen, we give a construction of the homoclinic and heteroclinic \(C^\ast\)-algebras of a shift space in terms of Bratteli diagrams. Lastly we present several specific examples which demonstrate these techniques. For the even shift we give a complete computation of all the associated invariants. We also present an example of a strictly non-sofic synchronizing shift. In particular we discuss the rank of the \(K\)-theory of the homoclinic algebra of a shift space and its implications. We also give a construction for producing from any minimal shift a synchronizing shift whose set of non-synchronizing points is exactly the original minimal shift.
ISSN:2331-8422