A holomorphic motion with non-subharmonic Hausdorff dimension

Abstract

We present an example of a holomorphic motion of a Cantor set $\lambda \mapsto A_{\lambda }$ for which neither the Hausdorff dimension $\dimH(A_λ)$ nor the lower Minkowski dimension $\dimMl(A_λ)$ is a subharmonic function of~$\lambda$. This answers negatively a question of Fuhrer--Ransford--Younsi, and distinguishes these dimensions from packing and upper Minkowski dimensions, for which subharmonicity holds. All four dimensions are computed explicitly.

10 pages