A class of Tribin functions related to $s$-symbol encodings of numbers with a zero redundancy

Abstract

In this paper, we consider a continuum class of continuous nowhere monotonic functions that generalize certain non-differentiable functions, including the Bush function, Wunderlich function, continuous Cantor projectors, Tribin function, etc. We consider a construction of the function related to $s$-symbol representations of numbers with a zero redundancy that are topologically equivalent to the classical $s$-adic representation (a value of the function has a two-symbol representation). Moreover, the condition on the first digit of a representation for the value of the function is more general than conditions considered before. The main object of study is a continuous function defined by equality \begin{gather*} f(Δ^{s^*}_{α_1α_2\ldotsα_n\ldots}) = Δ^{2^*}_{β_1β_2\ldotsβ_n\ldots}, \quad α_n \in \{ 0, 1, 2, \ldots, s - 1 \} \equiv A_s, β_1 = \begin{cases} 0 & \text{if ${\alpha }_1\in A_0$}, 1 & \text{if ${\alpha }_1\in A_1$}, \end{cases} \quad β_{n+1} = \begin{cases} β_n & \text{if ${\alpha }_{n+1}={\alpha }_n$}, 1 - β_n & \text{if ${\alpha }_{n+1}\ne {\alpha }_n$}. \end{cases} \end{gather*} where ${\Delta }_{{\alpha }_1{\alpha }_2\dots {\alpha }_n\dots }^{s^{\ast }}$ is an $s$-symbol representation of a number $x\in [0,1]$ that is topologically equivalent to the classical $s$-adic representation, ${\Delta }_{{\beta }_1{\beta }_2\dots {\beta }_n\dots }^{2^{\ast }}$ is a two-symbol representation that is topologically equivalent to the classical binary representation, and $A_0\cup A_1=A_s$, $A_0\ne A_s\ne A_1$.