A multifractal analysis of a singular potential and its equilibrium measure
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Potentials with logarithmic singularities occur naturally in the study of dynamical systems, for example, in systems with a zero of the derivative. They also appear in some number theoretical problems or in the study of substitution dynamical systems. We consider Thue-Morse (TM) measure which we regard as the (unique) equilibrium measure of a potential with logarithmic singularity. We study the uncommon fast scaling of the Birkhoff sums of the potential and the TM-measure by conducting multifractal analysis, using tools from thermodynamic formalism, which however has to be adapted to an unbounded potential. This is joint work with Ph. Gohlke and J. Schmeling.