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

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<jats:title>Abstract</jats:title><jats:p>The Sierpinski gasket <jats:italic>K</jats:italic> 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 <jats:italic>I</jats:italic>, is removed from <jats:italic>K</jats:italic>. At a glance, “the Sierpinski gasket minus the bottom line” <jats:inline-formula><jats:alternatives><jats:tex-math>$$K\backslash I$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>\</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> has a structure of a tree of Sierpinski gaskets. This observation leads us to the results showing that the boundary of <jats:inline-formula><jats:alternatives><jats:tex-math>$$K\backslash I$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>\</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is not the line segment <jats:italic>I</jats:italic> 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 <jats:italic>K</jats:italic> on the bottom line <jats:italic>I</jats:italic>.</jats:p>

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