Mini-Workshop: The Pisot Conjecture — From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets

2009 ◽  
pp. 725-766
Author(s):  
Valerie Berthe ◽  
David Damanik ◽  
Daniel Lenz
Keyword(s):  
Dynamical Systems ◽  
Substitution Dynamical Systems

scholarly journals Substitution dynamical systems on infinite alphabets

2006 ◽  
Vol 56 (7) ◽  
pp. 2315-2343 ◽  
Author(s):  
Sébastien Ferenczi
Keyword(s):  
Dynamical Systems ◽  
Substitution Dynamical Systems

scholarly journals Self-affine tiling via substitution dynamical systems and Rauzy fractals

2002 ◽  
Vol 206 (2) ◽  
pp. 465-485 ◽  
Author(s):  
Víctor F. Sirvent ◽  
Yang Wang
Keyword(s):  
Dynamical Systems ◽  
Substitution Dynamical Systems

scholarly journals Invariant measures on stationary Bratteli diagrams

2009 ◽  
Vol 30 (4) ◽  
pp. 973-1007 ◽  
Author(s):  
S. BEZUGLYI ◽  
J. KWIATKOWSKI ◽  
K. MEDYNETS ◽  
B. SOLOMYAK
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Complex Number ◽  
Incidence Matrix ◽  
Invariant Measures ◽  
Explicit Description ◽  
Bratteli Diagram ◽  
Probability Measures ◽  
Bratteli Diagrams ◽  
Substitution Dynamical Systems

AbstractWe study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

scholarly journals Coding of substitution dynamical systems as shifts of finite type

2014 ◽  
Vol 36 (3) ◽  
pp. 944-972 ◽  
Author(s):  
PAUL SURER
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Periodic Points ◽  
Basic Properties ◽  
Substitution Dynamical Systems ◽  
Shift Map ◽  
Special Case

We develop a theory that allows us to code dynamical systems induced by primitive substitutions continuously as shifts of finite type in many different ways. The well-known prefix–suffix coding turns out to correspond to one special case. We precisely analyse the basic properties of these codings (injectivity, coding of the periodic points, properties of the presentation graph, interaction with the shift map). A lot of examples illustrate the theory and show that, depending on the particular coding, several amazing effects may occur. The results give new insights into the theory of substitution dynamical systems and might serve as a powerful tool for further researches.

scholarly journals Aperiodic substitution systems and their Bratteli diagrams

2009 ◽  
Vol 29 (1) ◽  
pp. 37-72 ◽  
Author(s):  
S. BEZUGLYI ◽  
J. KWIATKOWSKI ◽  
K. MEDYNETS
Keyword(s):  
Dynamical Systems ◽  
Finite Rank ◽  
Bratteli Diagram ◽  
Bratteli Diagrams ◽  
Substitution Dynamical Systems ◽  
Substitution System ◽  
Minimal Component

AbstractWe 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.

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