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Orbit equivalent substitution dynamical systems and complexity


Authors: S. Bezuglyi and O. Karpel
Journal: Proc. Amer. Math. Soc. 142 (2014), 4155-4169
MSC (2010): Primary 37B10; Secondary 37A20, 37B05
DOI: https://doi.org/10.1090/S0002-9939-2014-12139-3
Published electronically: July 31, 2014
MathSciNet review: 3266986
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Abstract: For any primitive proper substitution $ \sigma $, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems $ \{(X_{\zeta _n}, T_{\zeta _n})\}_{n=1}^{\infty }$ such that they all are (strong) orbit equivalent to $ (X_{\sigma }, T_{\sigma })$. We show that the complexity of the substitution dynamical systems $ \{(X_{\zeta _n}, T_{\zeta _n})\}$ is the essential difference that prevents them from being isomorphic. Given a primitive (not necessarily proper) substitution $ \tau $, we find a stationary simple properly ordered Bratteli diagram with the least possible number of vertices such that the corresponding Bratteli-Vershik system is orbit equivalent to $ (X_{\tau }, T_{\tau })$.


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Additional Information

S. Bezuglyi
Affiliation: Institute for Low Temperature Physics, Kharkov, Ukraine
Email: bezuglyi@ilt.kharkov.ua

O. Karpel
Affiliation: Institute for Low Temperature Physics, Kharkov, Ukraine
Email: helen.karpel@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12139-3
Received by editor(s): January 10, 2012
Received by editor(s) in revised form: September 18, 2012, and January 3, 2013
Published electronically: July 31, 2014
Communicated by: Bryna Kra
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.