On the 𝐺-isomorphism of probability and dimensional theories of representations of real numbers and fractal faithfulness of systems of coverings

Garko, I.; Nikiforov, R.; Torbin, G.

doi:10.1090/tpms/1006

On the $G$-isomorphism of probability and dimensional theories of representations of real numbers and fractal faithfulness of systems of coverings

Authors: I. I. Garko, R. O. Nikiforov and G. M. Torbin
Translated by: S. Kvasko
Journal: Theor. Probability and Math. Statist. 94 (2017), 17-36
MSC (2010): Primary 11K55, 26A30, 28A80, 60G30
DOI: https://doi.org/10.1090/tpms/1006
Published electronically: August 25, 2017
MathSciNet review: 3553451
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new method is developed to construct the probabilistic and dimensional theories for families of representations of real numbers based on studies of special mappings that preserve the Lebesgue measure and Hausdorff–Besicovitch dimension. These mappings are characterized by the property that a preimage and image have the same symbols for two representations of the same family (the set of points of discontinuity of such mappings can be everywhere dense). These mappings are said to be $G$-mappings ($G$-isomorphisms of representations). The probabilistic, metric, and dimensional theories of $G$-isomorphic representations are identical. We establish a rather deep connection between the faithfulness of systems of coverings generated by different representations and the property of preservation of the Hausdorff-Besicovitch dimension of sets by the above-mentioned mappings. General sufficient conditions on faithfulness are found to evaluate the Hausdorff–Besicovitch dimension of families of cylinder sets generated by $F$ and $I$-$F$-representations of real numbers.

References

Sergio Albeverio, Yuri Kondratiev, Roman Nikiforov, and Grygoriy Torbin, On fractal properties of non-normal numbers with respect to Rényi $f$-expansions generated by piecewise linear functions, Bull. Sci. Math. 138 (2014), no. 3, 440–455 (English, with English and French summaries). MR 3206478, DOI https://doi.org/10.1016/j.bulsci.2013.10.005
Sergio Albeverio, Oleksandr Baranovskyi, Mykola Pratsiovytyi, and Grygoriy Torbin, The Ostrogradsky series and related Cantor-like sets, Acta Arith. 130 (2007), no. 3, 215–230. MR 2365703, DOI https://doi.org/10.4064/aa130-3-2
Sergio Albeverio, Iryna Garko, Muslem Ibragim, and Grygoriy Torbin, Non-normal numbers: full Hausdorff dimensionality vs zero dimensionality, Bull. Sci. Math. 141 (2017), no. 2, 1–19. MR 3614114, DOI https://doi.org/10.1016/j.bulsci.2016.04.001

S. Albeverio, Yu. Kondratiev, R. Nikiforov, and G. Torbin, On new fractal phenomena connected with infinite linear IFS (submitted to Math. Nachr., arXiv:1507.05672)

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, and G. Torbin, On fine structure of singularly continuous probability measures and random variables with independent $\~Q$-symbols, Methods Funct. Anal. Topology 17 (2011), no. 2, 97–111. MR 2849470

S. Albeverio, G. Ivanenko, M. Lebid, and G. Torbin, On the Hausdorff dimension faithfulness and the Cantor series expansion (submitted for publication in Math. Research Letters, arXiv:1305.6036)

Sergio Albeverio, Yuliya Kulyba, Mykola Pratsiovytyi, and Grygoriy Torbin, On singularity and fine spectral structure of random continued fractions, Math. Nachr. 288 (2015), no. 16, 1803–1813. MR 3417870, DOI https://doi.org/10.1002/mana.201500045

Sergio Albeverio, Mykola Pratsiovytyi, and Grygoriy Torbin, Transformations preserving the Hausdorff-Besicovitch dimension, Cent. Eur. J. Math. 6 (2008), no. 1, 119–128. MR 2379954, DOI https://doi.org/10.2478/s11533-008-0007-y

S. Albeverio, M. Pratsiovytyi, and G. Torbin, Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their $s$-adic digits, Ukraïn. Mat. Zh. 57 (2005), no. 9, 1163–1170 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 57 (2005), no. 9, 1361–1370. MR 2216038, DOI https://doi.org/10.1007/s11253-006-0001-0

Sergio Albeverio and Grygoriy Torbin, Fractal properties of singular probability distributions with independent $Q^*$-digits, Bull. Sci. Math. 129 (2005), no. 4, 356–367 (English, with English and French summaries). MR 2134126, DOI https://doi.org/10.1016/j.bulsci.2004.12.001

O. Baranov’kyĭ, M. Pratsiovytyĭ, and B. Get’man, Comparison of metric theories of representations of numbers by Engel and Ostrogradskyĭ and continued fractions, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 12 (2011), 130–139. (Ukrainian)

O. Baranov’kyĭ, M. Pratsiovytyĭ, and G. Torbin, Ostrogradskyĭ–Sierpiński–Pierce Series and Their Applications, “Naukova dumka”, Kyiv, 2013. (Ukrainian)

M. P. Bernardi and C. Bondioli, On some dimension problems for self-affine fractals, Z. Anal. Anwendungen 18 (1999), no. 3, 733–751. MR 1718162, DOI https://doi.org/10.4171/ZAA/909

Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027

C. D. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on ${\bf R}$, Internat. J. Math. Math. Sci. 11 (1988), no. 4, 643–649. MR 959443, DOI https://doi.org/10.1155/S016117128800078X

Karma Dajani and Cor Kraaikamp, Ergodic theory of numbers, Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, 2002. MR 1917322

János Galambos, Representations of real numbers by infinite series, Lecture Notes in Mathematics, Vol. 502, Springer-Verlag, Berlin-New York, 1976. MR 0568141

I. Garko, On a new approach to the study of fractal properties of probability measures with independent $x$-$Q_\infty$-digits. (submitted to Theory Stoch. Process.)

I. Garko, R. Nikiforov, and G. Torbin, $G$-isomorphism of number systems and faithfulness of covering systems. I, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 16 (2014), no. 1, 120–133. (Ukrainian)

I. Garko, R. Nikiforov, and G. Torbin, $G$-isomorphism of number systems and faithfulness of covering systems. II, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 16 (2014), no 2, 6–17. (Ukrainian)

B. Get’man, Metric properties of the set of number determined by assumptions imposed on their expansions in Engel series, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 10 (2009), 88–99. (Ukrainian)

Yu. Zhihareva and M. Pratsiovytyĭ, Representations of numbers by Łüroth series with positive terms: foundations of metric theory, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 9 (2008), 200–211. (Ukrainian)

Yu. Kondratiev, M. Lebid, O. Slutskyi, and G. Torbin, Cantor series expansions and packing dimension faithfulness. (submitted to Adv. Math., arXiv:1507.05663)

R. Nikiforov and G. Torbin, Ergodic properties of $Q_{\infty }$-representations and fractal properties of probability measures with independent $Q_\infty$-symbols, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 9 (2008), 80–103. (Ukrainian)

R. O. Nīkīforov and G. M. Torbīn, Fractal properties of random variables with independent $Q_\infty $-symbols, Teor. Ĭmovīr. Mat. Stat. 86 (2011), 150–162 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 86 (2013), 169–182. MR 2986457, DOI https://doi.org/10.1090/S0094-9000-2013-00896-5

R. Nikiforov and G. Torbin, On the Hausdorff–Besicovitch dimension of generalized self-similar sets generated by infinite IFS, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 13 (2012), 151–162. (Ukrainian)

Y. Peres and G. Torbin, Continued fractions and dimensional gaps. (in preparation)

I. Pratsevyta, Expansions of real numbers in second type Ostrogradskuĭ series ($O_2$- and $\bar {O_2}$-representations), their geometry and applications, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 9 (2008), 128–147. (Ukrainian)

M. Pratsiovytyĭ and M. Zadnipryanyĭ, Geometry and foundations of metric theory of representations of real numbers by Sylvester series, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 11 (2011), 76–85. (Ukrainian)

M. Pratsiovytyĭ and G. Torbin, On the classification of one dimensional singularly continuous probability measures according to their spectral properties, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 7 (2006), 140–151. (Ukrainian)

M. Pratsiovytyĭ and G. Torbin, Analytic (symbolic) representation of continuous transformations of $\mathbb R^1$ that preserve the Hausdorff–Besicovitch dimension, Sci. Bull. Dragomanov Nat. Pedagogical. Univer. Ser. 1. Phys. Mat. 4 (2003), 207–205. (Ukrainian)

M. Pratsiovytyĭ, Fractal Approach in Studies of Singular Distributions, M. P. Dragomanov Nat. Pedagogical. Univer. Publ., Kyiv, 1998. (Ukrainian)

C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862

Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1419320

G. M. Torbīn, On the DP-properties of fractal probability measures with independent $Q$-symbols, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 4 (2008), 44–50 (Ukrainian, with English summary). MR 2485212

G. M. Torbīn, Multifractal analysis of singularly continuous probability measures, Ukraïn. Mat. Zh. 57 (2005), no. 5, 706–721 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 57 (2005), no. 5, 837–857. MR 2209816, DOI https://doi.org/10.1007/s11253-005-0233-4

G. Torbin, Probabilistic approach to transformations preserving fractal dimension, Mat. Bull. Scientific Taras Shevchenko Soc. 4 (2007), no. 4, 275–283. (Ukrainian)

Grygoriy Torbin, Probability distributions with independent $Q$-symbols and transformations preserving the Hausdorff dimension, Theory Stoch. Process. 13 (2007), no. 1-2, 281–293. MR 2343830

A. F. Turbin and N. V. Pratsevityĭ, Fraktal′nye mnozhestva, funktsii, raspredeleniya, “Naukova Dumka”, Kiev, 1992 (Russian, with Russian and Ukrainian summaries). MR 1353239