“The Sierpinski gasket minus its bottom line” as a tree of Sierpinski gaskets

Abstract

The Sierpinski gasket K has three line segments constituting a regular triangle as its border. This paper studies what will happen if one of them, which is called the bottom line and is denoted by I, is removed from K. At a glance, “the Sierpinski gasket minus the bottom line” K\\I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\backslash I$$\\end{document} has a structure of a tree of Sierpinski gaskets. This observation leads us to the results showing that the boundary of K\\I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\backslash I$$\\end{document} is not the line segment I but a Cantor set from viewpoints of geometry and analysis. As a by-product, we have an explicit expression of the jump kernel of the trace of the Brownian motion of K on the bottom line I.