Spread of Values of a Cantor-Type Fractal Continuous Nonmonotone Function

Abstract

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 ‖q_ik‖, i ∈ A₅, k ∈ N, with positive elements (q_0k + q_1k + q_2k + q_3k + q_4k = 1) such that \( {\prod}_{k=1}^{\infty}\underset{i}{\max}\left\{{q}_{ik}\right\}=0 \) and β_0k = 0, β_{i + 1, k} = β_ik + q_ik, \( i=\overline{0,4} \), 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 δ_0n = 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., δ_{i + 1, n} = δ_in + g_in, n ∈ N, and (ε_n) is a given sequence of real numbers such that 0 ≤ ε_n ≤ 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.