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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
##### Keywords:
Morse sequence; self-joinings
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