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

TL;DR

This paper introduces a broad class of continuous, nowhere monotonic functions related to $s$-symbol number representations with zero redundancy, generalizing known functions like the Bush and Wunderlich functions, and explores their topological and functional properties.

Contribution

It constructs a new class of functions based on $s$-symbol representations with generalized digit conditions, extending previous models and analyzing their topological and functional characteristics.

Findings

The functions are continuous and nowhere monotonic.

They generalize several known non-differentiable functions.

The functions are topologically equivalent to classical $s$-adic and binary representations.

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(\Delta^{s^*}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}) = \Delta^{2^*}_{\beta_1\beta_2\ldots\beta_n\ldots}, \quad \alpha_n \in \{ 0, 1, 2, \ldots, s - 1 \} \equiv A_s, \beta_1 = \begin{cases} 0 & \text{if $\alpha_1 \in A_0$}, 1 & \text{if $\alpha_1 \in A_1$}, \end{cases} \quad \beta_{n+1} = \begin{cases} \beta_n & \text{if $\alpha_{n+1} = \alpha_n$}, 1 - \beta_n & \text{if $\alpha_{n+1} \neq \alpha_n$}. \end{cases} \end{gather*} where $\Delta^{s^*}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}$ is an $s$-symbol representation of a number $x \in [0, 1]$ that is topologically equivalent to the classical $s$-adic representation, $\Delta^{2^*}_{\beta_1\beta_2\ldots\beta_n\ldots}$ is a two-symbol representation that is topologically equivalent to the classical binary representation, and $A_0 \cup A_1 = A_s$, $A_0 \neq A_s \neq A_1$.