On new fractal phenomena connected with infinite linear IFS

Albeverio S, Kondratiev Y, Nikiforov R, Torbin G (2017)
MATHEMATISCHE NACHRICHTEN 290(8-9): 1163-1176.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Albeverio, SergioUniBi; Kondratiev, YuriUniBi; Nikiforov, Roman; Torbin, Grygoriy
Abstract / Bemerkung
We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Luroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non-faithfullness of the family of cylinders generated by the classical Luroth expansion. We also develop new approach to the study of subsets of Q8-essentially non-normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in [2] and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Luroth expansion. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Stichworte
Infinite IFS; Luroth expansion; Q(infinity)-expansion; Hausdorff; dimension; faithful and non-faithful nets; fractals; singular; probability measures; non-normal numbers
Erscheinungsjahr
2017
Zeitschriftentitel
MATHEMATISCHE NACHRICHTEN
Band
290
Ausgabe
8-9
Seite(n)
1163-1176
ISSN
0025-584X
eISSN
1522-2616
Page URI
https://pub.uni-bielefeld.de/record/2912338

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Albeverio S, Kondratiev Y, Nikiforov R, Torbin G. On new fractal phenomena connected with infinite linear IFS. MATHEMATISCHE NACHRICHTEN. 2017;290(8-9):1163-1176.
Albeverio, S., Kondratiev, Y., Nikiforov, R., & Torbin, G. (2017). On new fractal phenomena connected with infinite linear IFS. MATHEMATISCHE NACHRICHTEN, 290(8-9), 1163-1176. doi:10.1002/mana.201500471
Albeverio, S., Kondratiev, Y., Nikiforov, R., and Torbin, G. (2017). On new fractal phenomena connected with infinite linear IFS. MATHEMATISCHE NACHRICHTEN 290, 1163-1176.
Albeverio, S., et al., 2017. On new fractal phenomena connected with infinite linear IFS. MATHEMATISCHE NACHRICHTEN, 290(8-9), p 1163-1176.
S. Albeverio, et al., “On new fractal phenomena connected with infinite linear IFS”, MATHEMATISCHE NACHRICHTEN, vol. 290, 2017, pp. 1163-1176.
Albeverio, S., Kondratiev, Y., Nikiforov, R., Torbin, G.: On new fractal phenomena connected with infinite linear IFS. MATHEMATISCHE NACHRICHTEN. 290, 1163-1176 (2017).
Albeverio, Sergio, Kondratiev, Yuri, Nikiforov, Roman, and Torbin, Grygoriy. “On new fractal phenomena connected with infinite linear IFS”. MATHEMATISCHE NACHRICHTEN 290.8-9 (2017): 1163-1176.