The Dimension Spectrum of Some Dynamical Systems

We analyze the dimension spectrum previously introduced and measured experimentally by Jensen, Kadanoff, and Libchaber. Using large-deviation theory, we prove, for some invariant measures of expanding Markov maps, that the Hausdorff dimension f(~) of the set on which the measure has a singularity is a well-defined, concave, and regular function. In particular, we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number. We also show in these particular cases that the function f is universal.