Variation conjectures for nested fractals

Let $X$ be a nested fractal, let $BV(X)$ denote the BV space developed in the paper, and let a dendrite be a nested fractal whose points are joined by unique geodesic arcs. A continuous function is said to have finite variation when it belongs to $BV(X)$, and its classical distributional derivative is a finite Radon measure when it admits such a derivative.

Nested-fractal variation conjectures.

\begin{enumerate} \item \text{On a nested fractal that is a dendrite, such as the Vicsek set, the BV space can be completely described using an analogue of Stieltjes integration along geodesics, and hence BV functions may be described as having classical distributional derivatives that are finite Radon measures.} \item \text{On the Sierpinski gasket, and certain other nested fractals, any non-constant continuous function is of infinite variation.} \end{enumerate}

These claims propose a substantially more explicit description of BV functions on dendritic fractals and a strong obstruction to finite variation for non-constant continuous functions on the Sierpinski gasket and related examples.