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The paper summarizes properties of topological and sequence en- tropy of the Morse shift XM generated by the Thue-Morse sequence tM. The first part is an estimation of growth rate of possible subwords in tM. We show a polynomial upper bound on the number of finite subwords occuring in tM which is Cn2log3 for some constant C > 0. In the second part we prove that the sequence entropy of XM is achieved for the sequence r(i) = 22i − 1.
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