  Oscillatory critical amplitudes have been repeatedly observed in hierarchical
models and, in the cases that have been taken into consideration, these
oscillations are so small to be hardly detectable. Hierarchical models are
tightly related to iteration of maps and, in fact, very similar phenomena have
been repeatedly reported in many fields of mathematics, like combinatorial
evaluations and discrete branching processes. It is precisely in the context of
branching processes with bounded off-spring that T. Harris, in 1948, first set
forth the possibility that the logarithm of the moment generating function of
the rescaled population size, in the super-critical regime, does not grow near
infinity as a power, but it has an oscillatory prefactor. These oscillations
have been observed numerically only much later and, while the origin is clearly
tied to the discrete character of the iteration, the amplitude size is not so
well understood. The purpose of this note is to reconsider the issue for
hierarchical models and in what is arguably the most elementary setting -- the
pinning model -- that actually just boils down to iteration of polynomial maps
(and, notably, quadratic maps). In this note we show that the oscillatory
critical amplitude for pinning models and the oscillating pre factor connected
to the Harris random variable coincide. Moreover we make explicit the link
between these oscillatory functions and the geometry of the Julia set of the
map, making thus rigorous and quantitative some ideas set forth by B. Derrida,
C. Itzykson and J. M. Luck (CMP 1984).
