Linearly recurrent systems

A non-erasing morphism σ is said to be recognizable on x if there exists ` such that, for each m ∈ Cσ(x), m′ ∈ Z, y[m−`,m+`) = y[m′−`,m′+`) implies that m′ ∈ Cσ(x), with y = σ(x). Is the Thue-Morse substitution σ : a 7→ ab, b 7→ ba recognizable on σ∞(a)? Same question for the Fibonacci substitution τ : a 7→ ab, b 7→ a. Let σ be a primitive morphism that is recognizable on some x and injective on letters. Let X = O(x) and Y = ∪k∈ZSσ(X), where S is the shift map. Prove that the set σ(X) is a clopen subset of Y , that the map σ : X → σ(X) is a homeomorphism, and that the collection P = {Sσ([a]) : a ∈ A, 0 ≤ k < |σ(a)|} is a clopen partition of Y . Let us note that recognizability has been proved by Mossé (1992) for primitive substitutions, and by Bezuglyi, Kwiatkowski and Medynets (2009) for any aperiodic substitution. What can be said for the substitution σ : 0 7→ 010, 1 7→ 10 in the case of one-sided words in AN and one-side recognizability? ∗Many thanks to S. Barbieri for his careful reading and suggestions.