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 $α_1 \in A_0$}, 1 & \text{if $α_1 \in A_1$}, \end{cases} \quad β_{n+1} = \begin{cases} β_n & \text{if $α_{n+1} = α_n$}, 1 - β_n & \text{if $α_{n+1} \neq α_n$}. \end{cases} \end{gather*} where $Δ^{s^*}_{α_1α_2\ldotsα_n\ldots}$ is an $s$-symbol representation of a number $x \in [0, 1]$ that is topologically equivalent to the classical $s$-adic representation, $Δ^{2^*}_{β_1β_2\ldotsβ_n\ldots}$ is a two-symbol representation that is topologically equivalent to the classical binary representation, and $A_0 \cup A_1 = A_s$, $A_0 \neq A_s \neq A_1$.