On bivariate Archimedean copulas with fractal support

Abstract Due to their simple analytic form (bivariate) Archimedean copulas are usually viewed as very smooth and handy objects, which should distribute mass in a fairly regular and certainly not in a pathological way. Building upon recently established results on the Archimedean family and working with iterated function systems with probabilities, we falsify this natural conjecture and derive the surprising result that for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> <m:mspace width="0.33em"/> <m:mo>∈</m:mo> </m:math> s\hspace{0.33em}\in [1, 2] there exists some bivariate Archimedean copula <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msub> </m:math> {A}_{s} fulfilling that the Hausdorff dimension of the support of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msub> </m:math> {A}_{s} is exactly <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s .