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

Mykola Pratsiovytyi, Sofiia Ratushniak, Oleksandr Baranovskyi, Iryna Lysenko

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 $A_0 \cup A_1 = A_s$, $A_0 \neq A_s \neq A_1$.

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 -symbol representations of numbers with a zero redundancy that are topologically equivalent to the classical -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 }, 1 & \text{if }, \end{cases} \quad β_{n+1} = \begin{cases} β_n & \text{if }, 1 - β_n & \text{if }. \end{cases} \end{gather*} where is an -symbol representation of a number that is topologically equivalent to the classical -adic representation, is a two-symbol representation that is topologically equivalent to the classical binary representation, and , .
Paper Structure (7 sections, 7 theorems, 35 equations)

This paper contains 7 sections, 7 theorems, 35 equations.

Table of Contents

  1. An encoding of real numbers and their $g$-representation
  2. The main object of study
  3. Well-definedness of the function and its continuity
  4. Range and level sets of the function
  5. Nowhere monotonocity of the function
  6. Variation of the function
  7. Concluding remarks

Key Result

Lemma 1

Let $x_0$ be a $g$-unary point. Then the condition $x \to x_0$ is equivalent to the condition $m \to \infty$ where $\alpha_m(x) \neq \alpha_m(x_0)$ but $\alpha_i(x) = \alpha_i(x_0)$ for $i < m$.

Theorems & Definitions (17)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 7 more