Bounded Variation on the Sierpinski Gasket

TL;DR

This paper investigates the fractal dimensions of graphs of functions on the Sierpinski gasket, establishing bounds and properties of functions with bounded variation in this fractal setting.

Contribution

It introduces new definitions of bounded variation on the Sierpinski gasket and analyzes the fractal dimensions of their graphs, providing key bounds and properties.

Findings

Fractal dimension of continuous bounded variation functions is log 3/log 2.

Class of bounded variation functions is closed under arithmetic operations.

Functions of bounded variation are almost everywhere continuous in the Hausdorff measure sense.

Abstract

Under certain continuity conditions, we estimate upper and lower box dimension of graph of a function defined on the Sierpinski gasket. We also give an upper bound for Hausdorff dimension and box dimension of graph of function having finite energy. Further, we introduce two sets of definitions of bounded variation for a function defined on the Sierpinski gasket. We show that fractal dimension of graph of a continuous function of bounded variation is log 3/log 2. We also prove that the class of all bounded variation functions is closed under arithmetic operations. Furthermore, we show that every function of bounded variation is continuous almost everywhere in the sense of log 3/log 2 dimensional Hausdorff measure.