LIMITING CURVES FOR THE DYADIC ODOMETER

摘要

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↦Stln−tSln/Rn∈C01,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ tmapsto left(Sleft(t{l}_night)- tSleft({l}_night)ight)/{R}_nin Cleft(left[0,1ight]ight), $$end{document}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 fωi=∑i≥0ωiqi+1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ fleft(left({omega}_iight)ight)=sum limits_{ige 0}{omega}_i{q}^{i+1} $$end{document} 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.