A note on Hausdorff measures of quasi-self-similar sets

Abstract

Sullivan has demonstrated that quasi-self-similarity provides a useful point of view for the study of expanding dynamical systems. In [ <bold>4</bold> , p. 57] he posed the question: Is the Hausdorff measure of a quasi-self-similar set positive and finite in its Hausdorff dimension? This paper answers both parts of this question. In <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="section-sign 1"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> § </mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\S 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the positivity is established for compact sets, and a lower bound is given for their Hausdorff measure. However, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="section-sign 2"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> § </mml:mi> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\S 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the finiteness is disproved. In fact, a quasi-self-similar set is constructed for which the Hausdorff measure is actually <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -infinite.