Limiting Curves for the Dyadic Odometer

A limiting curve of a stationary process in discrete time was defined by É. Janvresse, T. de la Rue, and Y. Velenik as the uniform limit of the functionst↦(S(tln)−tS(ln))/Rn∈C([01]), where S stands for the piecewise linear extension of the partial sum, Rn:= sup |S(tln) − tS(ln))|, and (ln) = (ln(ω)) is a suitable sequence of integers. We determine the limiting curves for the stationary sequence (f ∘ Tn(ω)) where T is the dyadic odometer on {0, 1}ℕ and f((ωi))=∑i≥0ωiqi+1 for 1/2 < |q| < 1. Namely, we prove that for a.e. ω there exists a sequence (ln(ω)) such that the limiting curve exists and is equal to (−1) times the Tagaki–Landsberg function with parameter 1/2q. The result can be obtained as a corollary of a generalization of the Trollope–Delange formula to the q-weighted case. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.