There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets.

Department of Mathematics University of Illinois at Urbana-Champaign

We discuss extremal problems related to planar conformal geometry. More specifically, we discuss the multifractal properties of the harmonic measure and boundary rotation, in relation to the boundary behavior of conformal maps.

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets.

Department of Mathematics University of Illinois at Urbana-Champaign