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Generic points in the Cartesian powers of the Morse dynamical system. (English) Zbl 1055.37015
The Prouhet-Thue-Morse sequence (or simply Morse sequence) is the binary sequence \((u_n)_{n\geq 0}\) defined as follows: \(u_n= 1\) if there is an odd number of 1’s in the base 2 expansion of the integer \(n\) and \(u_n= 0\) otherwise. The symbolic dynamical system \({\mathcal M}\) naturally associated with the Morse sequence is known as one of the simplest example of a strictly ergodic dynamical system.
The authors deal with ergodic properties of some self-joinings of the Morse dynamical system. In particular, they discover quite surprising phenomena. Indeed, they prove that in the Cartesian square and cube of \({\mathcal M}\) every point is generic (with respect to a measure that depends on the point), whereas there are a lot of nongeneric points in the \(k\)th Cartesian power of \({\mathcal M}\), as soon as \(k\) is at least 4.
MSC:
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
37A25 Ergodicity, mixing, rates of mixing
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B10 Symbolic dynamics
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