Non-affine fractal hypersurfaces: construction and dimensions

TL;DR

This paper constructs non-affine fractal hypersurfaces on simplices and estimates their fractal dimensions and measures, advancing understanding of complex geometric structures in higher dimensions.

Contribution

It introduces a method to construct non-affine fractal hypersurfaces and provides new estimates for their fractal dimensions and invariant measures.

Findings

Fractal dimension of the graph of non-affine multivariate fractal functions is estimated.

Upper bounds for Hausdorff dimension of invariant measures are derived.

Construction techniques for non-affine fractal hypersurfaces are developed.

Abstract

This article presents the construction of a non-affine hypersurface on an $n$-simplex in $\mathbb{R}^n$. Additionally, fractal dimension of the graph of a non-affine multivariate real-valued fractal function is estimated under certain conditions. Furthermore, the upper bound of the Hausdorff dimension of the invariant probability measure supported on the graph of such fractal function is estimated.