Generalization of the Tribin Function

We consider the Tribin function and its generalization based on the ${Q}_s^{\ast }$ -representation of real numbers, which is an s-symbol encoding of numbers and, generally speaking, a nonself-similar generalization of the s-adic representation. By definition, the function f associates the number $x={\varDelta}_{\upalpha_1{\upalpha}_2\dots {\upalpha}_n\dots}^{Q_s^{\ast }},$ where α_n ∈ L ≡ A_s × A_s × … × A_s × … and A_s = {0, 1, …, s − 1} is an alphabet, s ≥ 3; with the number $y=f(x)={\varDelta}_{\upgamma_1{\upgamma}_2\dots {\upgamma}_n\dots}^{G_2^{\ast }},$

$$ {\gamma}_1=\left\{\begin{array}{l}0\kern1em \mathrm{for}\kern1em {\upalpha}_1=0,\\ {}1\kern1em \mathrm{for}\kern1em {\upalpha}_1\ne 0,\end{array}\right.{\gamma}_{n+1}=\left\{\begin{array}{l}{\gamma}_n\kern2.5em \mathrm{for}\kern1em {\upalpha}_{n+1}={\upalpha}_n,\\ {}1-{\gamma}_n\kern1em \mathrm{for}\kern1em {\upalpha}_{n+1}\ne {\upalpha}_n,\end{array}\right. $$

where the ${G}_2^{\ast }$ -representation of numbers has the two-symbol alphabet A₂ = {0, 1}. We prove that the function f is well-defined, continuous, and nowhere monotone. Its variational properties are also analyzed.