M. Urbanski B.O. Stratmann
Multifractal Analysis for Parabolically Semihyperbolic Generalized Polynomial-Like Map
Preprint series: Mathematica Gottingensis

MSC:

28A78 Hausdorff measures

Abstract: In this paper we study parabolically semihyperbolic
generalized polynomial-like maps and give a finer fractal analysis
of their Julia sets. We discuss various generalizations of the
classical notion of topological pressure to situations in which the
underlying potentials are not necessarily continuous or bounded.
Subsequently, we investigate various types of conformal measures and
invariant Gibbs states, which then enables us to deduce analytic
properties for the generalized pressure functions. On the basis of
these results, we finally derive our multifractal analysis, and then show
that for the special case in which the Julia set does not contain
critical points, this general multifractal analysis has a
more transparent geometric interpretation in terms of the local
scaling behaviour of the canonically associated equilibrium state.