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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.
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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