We present in this paper a new method to deal with automatic sequences. This method allows us to prove a Möbius-randomness-principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a Prime Number Theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to 0.

Abstract. We prove that the Thue-Morse sequence t along subsequences indexed by ⌊n c ⌋ is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence n → t ⌊n c ⌋ has asymptotic density 2 −L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.In particular, this result shows that for 1 < c < 3/2 the sequence n → t ⌊n c ⌋ attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue-Morse sequence, that is, it satisfies a Bombieri-Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} L is contained as an arithmetic subsequence of t.

We prove that automatic sequences generated by synchronizing automata satisfy the full Sarnak conjecture. This is of particular interest, since Berlinkov [2,3] proved recently that almost all automata are synchronizing.

We show that the (morphic) sequence (−1) sϕ(n) is asymptotically orthogonal to all bounded multiplicative functions, where s ϕ denotes the Zeckendorf sum-of-digits function. In particular we have n

In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from twisted compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a letter-to-letter code.1991 Mathematics Subject Classification. Primary 37B10, 11B85, Secondary 37B05, 37B15.