Aperiodic substitution systems and their Bratteli diagrams
By S. Bezuglyi, J. Kwiatkowski, Konstantin Medynets. Ergodic Theory and Dynamical Systems, 2007
84 citations recorded by the source corpus.
Abstract We study aperiodic substitution dynamical systems arising from non-primitive substitutions. We prove that the Vershik homeomorphism φ of a stationary ordered Bratteli diagram is topologically conjugate to an aperiodic substitution system if and only if no restriction of φ to a minimal component is conjugate to an odometer. We also show that every aperiodic substitution system generated by a substitution with nesting property is conjugate to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitution system is recognizable. The classes of m-primitive substitutions and derivative substitutions associated with them are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.
Subfields: Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/28dd5904cf573da94e2c089bacfdaac6da5e3590
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In Computational Theory and Mathematics
- S. Bezuglyi
- J. Kwiatkowski
- Boris Solomyak
- Konstantin Medynets
- Val'erie Berth'e
- Vincent Delecroix
- Pablo Shmerkin
Citations in the corpus
- Finite rank Bratteli diagrams: Structure of invariant measures (2010, background): “X max and X min the sets of all maximal and minimal infinite paths from XB, respectively. It is not hard to see that X max and X min are finite sets for finite rank Bratteli diagrams (Proposition 6.2 in [BKM09]). Let X∗ B be the E-invariant set of all infinite paths which are cofinal neither to a maximal”
- Invariant measures on stationary Bratteli diagrams (2008, methodology): “t B= (V,E,ω) be a stationary ordered Bratteli diagram, or more generally, suppose N= sup n|V | <∞. Then it is easy to see that the sets X max(ω) and X min(ω) of maximal and minimal paths are finite [BKM]. Indeed, observe that two maximal paths which go through the same vertex at level nmust have t”
- Beyond substitutive dynamical systems: S-adic expansions (2013, background): “For more on the connections between Vershik adic systems and S‐adic ones, see [37], \lfloor 38 , Sec.”