Transformations of $(0,1]$ preserving tails of~$\Delta^{\mu}$-representation of numbers

Tetiana M. Isaieva ; Mykola V. Pratsiovytyi

Algebra and discrete mathematics, Tome 22 (2016) no. 1, pp. 102-115.

Résumé

Let $\mu\in (0,1)$ be a given parameter, $\nu\equiv 1-\mu$. We consider $\Delta^{\mu}$-representation of numbers $x=\Delta^{\mu}_{a_1a_2\ldots a_n\ldots}$ belonging to $(0,1]$ based on their expansion in alternating series or finite sum in the form: $$ x=\sum_n(B_{n}-{B'_n})\equiv \Delta^{\mu}_{a_1a_2\ldots a_n\ldots}, $$ where $B_n=\nu^{a_1+a_3+\ldots+a_{2n-1}-1}{\mu}^{a_2+a_4+\ldots+a_{2n-2}}$, ${B^{\prime}_n}=\nu^{a_1+a_3+\ldots+a_{2n-1}-1}{\mu}^{a_2+a_4+\ldots+a_{2n}}$, $a_i\!\in\! \mathbb{N}$. This representation has an infinite alphabet $\{1,2,\ldots\}$, zero redundancy and $N$-self-similar geometry. In the paper, classes of continuous strictly increasing functions preserving “tails” of $\Delta^{\mu}$-representation of numbers are constructed. Using these functions we construct also continuous transformations of $(0,1]$. We prove that the set of all such transformations is infinite and forms non-commutative group together with an composition operation.

Détail

Keywords: $\Delta^{\mu}$-representation, cylinder, tail set, function preserving “tails” of $\Delta^{\mu}$-representation of numbers, continuous transformation of $(0,1]$ preserving “tails” of $\Delta^{\mu}$-representation of numbers
Mots-clés : group of transformations.

Bibliographie

[1] G. Alkauskas, “An asymptotic formula for the moments of the Minkowski question mark function in the interval $[0,1]$”, Lithuanian Mathematical Journal, 48:4 (2008), 357–367 | DOI | MR | Zbl

[2] G. Alkauskas, “Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function”, Involve, 2:2 (2009), 121–159 | DOI | MR | Zbl

[3] G. Alkauskas, “The Minkowski question mark function: explicit series for the dyadic period function and moments”, Mathematics of Computation, 79:269 (2010), 383–418 | DOI | MR | Zbl

[4] G. Alkauskas, “Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function”, Ramanujan J., 25:3 (2011), 359–367 | DOI | MR | Zbl

[5] G. Alkauskas, “The Minkowski $?(x)$ function and Salem's problem”, C.R. Acad. Sci. Paris, 350:3–4 (2012), 137–140 | DOI | MR | Zbl

[6] A. Denjoy, “Sur une fonction de Minkowski”, C. R. Acad. Sci., Paris, 194 (1932), 44–46

[7] A. Denjoy, “Sur une fonction réelle de Minkowski”, J. Math. Pures Appl., 17 (1938), 105–151

[8] A. A. Dushistova, I. D. Kan, N. G. Moshchevitin, “Differentiability of the Minkowski question mark function”, J. Math. Anal. Appl., 401:2 (2013), 774–794 | DOI | MR | Zbl

[9] A. Ya. Khinchin, Continued fractions, Nauka, Moscow, 1978, 116 pp. (Russian)

[10] J. R. Kinney, “Note on a singular function of Minkowski”, Proc. Amer. Math. Soc., 11:5 (1960), 788–794 | DOI | MR | Zbl

[11] M. Kesseböhmer, B. Stratmann, “Fractal analysis for sets of non-differentiability of Minkowski's question mark function”, J. Number Theory, 128:9 (2008), 2663–2686 | DOI | MR | Zbl

[12] M. Lamberger, “On a family of singular measures related to Minkowski's $?(x)$ function”, Indag. Math., 17:1 (2006), 45–63 | DOI | MR | Zbl

[13] H. Minkowski, Gesammelte Abhandlungen, v. 2, B. G. Teubner, Berlin, 1911 | Zbl

[14] O. R. Beaver, T. Garrity, “A two-dimensional Minkowski $?(x)$ function”, J. Number Theory, 107:1 (2004), 105–134 | DOI | MR | Zbl

[15] G. Panti, “Multidimensional continued fractions and a Minkowski function”, Monatsh. Math., 154:3 (2008), 247–264 | DOI | MR | Zbl

[16] J. Paradis, P. Viader, L. Bibiloni, “A new light on Minkowski's $?(x)$ function”, J. Number. Theory, 73:2 (1998), 212–227 | DOI | MR | Zbl

[17] M. V. Pratsiovytyi, Fractal approach to investigation of singular probability distributions, Natl. Pedagog. Mykhailo Drahomanov Univ. Publ., Kyiv, 1998, 296 pp. (Ukrainian)

[18] O. Baranovskyi, Yu. Kondratiev, M. Pratsiovytyi, Transformations and functions preserving tails of $E$-representation of numbers, Manuscript, 2015

[19] M. V. Pratsiovytyi, T. M. Isaieva, “$\Delta^{\mu}$-representation as a generalization of $\Delta^{\sharp}$-representation and a foundation of new metric theory of real numbers”, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Drahomanova. Ser. 1. Fiz.-Mat. Nauky, 16 (2014), 164–186 (Ukrainian)

[20] M. V. Pratsiovytyi, T. M. Isaieva, “Encoding of real numbers with infinite alphabet and base 2”, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Drahomanova. Ser. 1. Fiz.-Mat. Nauky, 15 (2013), 6–23 (Ukrainian)

[21] M. V. Pratsiovytyi, T. M. Isaieva, “On some applications of $\Delta^{\sharp}$-representation of real numbers”, Bukovyn. Mat. Zhurn., 2:2–3 (2014), 187–197 (Ukrainian)

[22] M. V. Pratsiovytyi, T. M. Isaieva, “Fractal functions related to $\Delta^{\mu}$-representation of numbers”, Bukovyn. Mat. Zhurn., 3:3–4 (2015), 156–165 (Ukrainian)

[23] M. V. Pratsiovytyi, A. V. Kalashnikov, “Singularity of functions of one-parameter class containing the Minkowski function”, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Drahomanova. Ser. 1. Fiz.-Mat. Nauky, 12 (2011), 59–65 (Ukrainian)

[24] M. V. Pratsiovytyi, A. V. Kalashnikov, V. K. Bezborodov, “On one class of singular functions containing classic Minkowski function”, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Drahomanova. Ser. 1. Fiz.-Mat. Nauky, 11 (2010), 207–213 (Ukrainian)

[25] R. Salem, “On some singular monotonic function which are strictly increasing”, Trans. Amer. Math. Soc., 53 (1943), 423–439 | DOI | MR