BOUNDED VARIATION ON THE SIERPINSKI GASKET

Under certain continuity conditions, we estimate upper and lower box dimensions of the graph of a function defined on the Sierpinski gasket. We also give an upper bound for Hausdorff dimension and box dimension of the graph of a 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 the 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.

Automated summary

In this study, we estimate the upper and lower box dimensions of the graph of a function defined on the Sierpinski gasket under certain continuity conditions. We also provide an upper bound for the Hausdorff dimension and box dimension of the graph of a function with finite energy. Furthermore, we introduce two sets of definitions of bounded variation for a function defined on the Sierpinski gasket and prove that the fractal dimension of the graph of a continuous function of bounded variation is log 3/log 2. We also demonstrate the closure of the class of all bounded variation functions under arithmetic operations and prove that every function of bounded variation is continuous almost everywhere according to the log 3/log 2-dimensional Hausdorff measure.