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Zapiski Nauchnykh Seminarov POMI, 2019, Volume 481, Pages 74–86
(Mi znsl6779)
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Limiting curves for the dyadic odometer
A. R. Minabutdinov National Research University Higher School of Economics, Department of Mathematics, St. Petersburg, Russia
Abstract:
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 functions
t↦(S(tln)−tS(ln))/Rn∈C([0,1]),
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.
Key words and phrases:
limiting curves, weighted sum-of-binary-digits function, Takagi–Landsberg curve, q-analogue of the Trollope–Delange formula.
Received: 19.09.2019
Citation:
A. R. Minabutdinov, “Limiting curves for the dyadic odometer”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Zap. Nauchn. Sem. POMI, 481, POMI, St. Petersburg, 2019, 74–86
Linking options:
https://www.mathnet.ru/eng/znsl6779 https://www.mathnet.ru/eng/znsl/v481/p74
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Abstract page: | 94 | Full-text PDF : | 19 | References: | 31 |
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