HD0L-$omega$-equivalence and periodicity problems in the primitive case (to the memory of G. Rauzy)

1. Introduction 1.1. Some words about Gérard Rauzy. I started my PhD in 1992 under the direction of G. Rauzy. The topic of my PhD was Substitutions and Bratteli diagrams. G. Rauzy was a worldwide recognized expert on substitutions (and many other topics) but Bratteli diagram representations of substitution subshifts was not one of these subjects. Thus I had an unformal second advisor, B. Host. Later, as a common agreement, he became my advisor. Some days after I begun my PhD he told me that he noticed something amazing about substitutions. He wrote on the blackboard the Fibonacci fixed point on the alphabet {0, 1} starting with 0. He considered all occurrences of the letter 0 and coded the words between two consecutive such occurrences. It is not surprising (see [Morse and Hedlund 1940]) that only two words appear : 01 and 0. If you respectively code them by 0 and 1 then you again obtain the same sequence. This is not surprising too because this sequence is the fixed point of the morphism defined by 0 → 01 and 1 → 0. Then he wrote a long prefix of the fixed point of the Morse sequence that we call x. Applying the same coding process (see below) to x it clearly appeared that the new sequence called the derived sequence of x, and denoted by D(x), is not the same for the obvious reason that 3 different letters occur. He repeated the process to this new sequence. Let's do it. The result was a sequence on a 4 letter alphabet. Once again he repeated the process. Amazingly we obtained the same sequence (the same long prefix). Repeating the process will clearly produce the same sequence. The process is periodic. Then he showed me another example where the period was 2. He was wondering if such a period always exists for fixed points of primitive substitutions. He knew that when this process is eventually periodic then the sequence is primitive substitutive and he explained me the proof on the blackboard. This proof was very useful for my PhD and for the present paper : this is the construction exposed in Proposition 17. It became a second subject in my PhD that I succeeded to solve in [Durand 1998a] and that provided me many arguments to obtain an extension of Cobham's theorem for primitive substitutions ( [Durand 1998b]).

When he showed me this process I found curious to look at it. Why was it interesting ? In fact, at this time, I did not know much about the work of G. Rauzy and almost nothing about interval exchange transformations (iets) and what he did with. I realized quickly that this process was in fact the induction process defined on dynamical systems by the Poincaré first return map : the derived sequence D(x) generates a subshift that is conjugate to the induced dynamical system (X, T ) on the cylinder set [0], where (X, T ) is the subshift generated by x. Thus, this corresponds to the induction process he defined [Rauzy 1979] to study iets. He induced on the left most interval and showed that the induced tranformation is again an iet with the same permutation. This is also the case for primitive substitutions : derived sequences of purely primitive substitutive sequences are purely primitive substitutive sequences ( [Durand 1998a]). His goal was to tackle the Keane’s conjecture saying that almost all iets are uniquely ergodic. This was solved independently in [Masur 1982] and [Veech 1982]. Later M. Boshernitzan and C. R. Carroll [Boshernitzan and Carroll 1997] proved that, for an iet defined on a quadratic field, “every consistent method of induction is eventually periodic” (up to rescaling). This is what I obtained in [Durand 1998a] in the context of primitive substitutions and that was expected by G. Rauzy. Thus I imagine the intuition of G. Rauzy was certainly leaded by iet considerations. I am very grateful to Gérard Rauzy for the wonderful subject he gave me and for the opportunity he offered me to do research.

1.2. The HD0L ω-equivalence problem. In this paper we propose to apply what the author developped in [Durand 1998a] and [Durand 1998b] (to answer the question of G. Rauzy) to solve the following problems in the primitive case. To describe these problems we need some notations. Let σ : A * → A * and τ : B * → B * be morphisms, and, u and v be words such that lim n→∞ σ n (u) and lim n→∞ τ n (v) converge to respectively x ∈ A N and y ∈ B N . We also need two morphisms φ :

This problem is open for more than 30 years and ask whether it is decidable to know if two morphic sequences are equal. More precisely, Is the equality “φ(x) = ψ(y)” decidable?

In this paper we answer positively to this problem for primitive morphisms. The equality problem “x = y” (also called the D0L ω-equivalence problem) was solved in 1984 by K. Culik II and T. Harju [Culik and Harju 1984]. An other proof was given by J. Honkala in [Honkala 2007] (see also [Honkala 2009a]).

1.2.2. More on the equality of sequences generated by morphisms. Once the equality “φ(x) = ψ(y)” or “x = y” is satisfied, it is natural to look fo

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