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On the Pisot substitution conjecture. (English) Zbl 1376.37043
Kellendonk, Johannes (ed.) et al., Mathematics of aperiodic order. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0902-3/hbk; 978-3-0348-0903-0/ebook). Progress in Mathematics 309, 33-72 (2015).
Summary: Our goal is to present a unified and reasonably complete account of the various conjectures, known as Pisot conjectures, that assert that certain dynamical systems arising from substitutions should have pure discrete dynamical spectrum. We describe the various contexts (symbolic, geometrical, arithmetical) in which substitution dynamical systems arise and review the relevant properties of these systems. The Pisot substitution conjecture is stated in each context and the relationships between these statements, and with several related conjectures, are discussed. We survey the special cases in which the Pisot substitution conjecture has been verified and present algorithmic procedures for checking pure discrete spectrum. We conclude with a discussion of possible extensions to higher dimensions.
For the entire collection see [Zbl 1338.37005].

MSC:
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B10 Symbolic dynamics
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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