Systems of numeration and fractal functions relating to substitutions (French)

Let A be a finite alphabet, σ a substitution over A, (un)n ϵ N a fixed point for σ, and for each a ϵ A, ƒ(a) a real number. We establish, under some assumptions, an asymptotic formula concerning the sum Sƒ (N) = Σi ⩽ N ƒ(ui), N ϵ N. This result generalizes some previous results from Coquet or Brillhart, Erdos, and Morton. Moreover, relations with self-affine functions (in a sense which generalizes a definition from Kamae) are proved. The calculi leave over systems of representation of integers and real numbers.