Abstract: The Sarnak conjecture states that any sequence obtained by a deterministic dynamical system is expected to be orthogonal to the Möbius function. Such results are often closely related to results about prime numbers and the Sarnak conjecture fuelled the interplay between dynamical systems and number theory.
We concentrate in this talk on sequences that are fixed under a substitution on a finite alphabet. The corresponding dynamical systems have been studied in a variety of contexts and the most prominent example is probably the Thue-Morse sequence which is the fixed point of the substitution 0->01, 1->10.

We will discuss two particular cases of substitutions:
1) Substitutions of constant length - which were handled by the author a few years ago.
2) A substitution corresponding to the Zeckendorf sum-of-digits function - which was handled recently by Drmota, Spiegelhofer and the author.