Spread of Values of a Cantor-Type Fractal Continuous Nonmonotone Function
| Autor: | M. V. Prats’ovytyi, O. V. Svynchuk |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Journal of Mathematical Sciences. 240:342-357 |
| ISSN: | 1573-8795 1072-3374 |
| DOI: | 10.1007/s10958-019-04356-0 |
| Popis: | By using the $$ {Q}_5^{\ast } $$ -representation of numbers $$ \left[0,1\right]\ni x={\beta}_{\alpha_1(x)1}+\sum \limits_{k=2}^{\infty}\left({\beta}_{\alpha_k(x)k}\prod \limits_{j=1}^{k-1}{q}_{\alpha_j(x)j}\right)={\Delta}_{\alpha_1(x){\alpha}_2(x)\dots {\alpha}_k(x)\dots}^{Q_5^{\ast }} $$ determined by the quinary alphabet A5 ≡ {0, 1, 2, 3, 4} and an infinite stochastic matrix ‖qik‖, i ∈ A5, k ∈ N, with positive elements (q0k + q1k + q2k + q3k + q4k = 1) such that $$ {\prod}_{k=1}^{\infty}\underset{i}{\max}\left\{{q}_{ik}\right\}=0 $$ and β0k = 0, βi + 1, k = βik + qik, $$ i=\overline{0,4} $$ , we define a continuous Cantor-type function by the equality $$ f\left({\Delta}_{\alpha_1\dots {\alpha}_k\dots}^{Q_5^{\ast }}\right)={\delta}_{\alpha_1(x)1}+\sum \limits_{k=2}^{\infty}\left({\delta}_{\alpha_k(x)k}\sum \limits_{j=1}^{k-1}{g}_{\alpha_j(x)j}\right)\equiv {\Delta}_{\alpha_1(x)\dots {\alpha}_k(x)\dots}^G, $$ where δ0n = 0, $$ {\delta}_{1n}=\frac{2+{\varepsilon}_n}{4} $$ , $$ {\delta}_{2n}=\frac{2}{4}={\delta}_{3n} $$ , and $$ {\delta}_{4n}=\frac{2-{\varepsilon}_n}{4} $$ , i.e., δi + 1, n = δin + gin, n ∈ N, and (en) is a given sequence of real numbers such that 0 ≤ en ≤ 1. We prove that this function is well defined and continuous. Moreover, it does not have intervals of monotonicity, except the intervals where it is constant. A criterion of bounded variation of the function is also established. We are especially interested in the problem of level sets of the function and in the topological and metric properties of the images of Cantor-type sets. |
| Databáze: | OpenAIRE |
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