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