  Since the pioneering works of Jakobson and Benedicks & Carleson and others,
it has been known that a positive measure set of quadratic maps admit invariant
probability measures absolutely continuous with respect to Lebesgue. These
measures allow one to statistically predict the asymptotic fate of Lebesgue
almost every initial condition. Estimating fluctuations of empirical
distributions before they settle to equilibrium requires a fairly good control
over large parts of the phase space. We use the sub-exponential slow recurrence
condition of Benedicks & Carleson to build induced Markov maps of arbitrarily
small scale and associated towers, to which the absolutely continuous measures
can be lifted. These various lifts together enable us to obtain a control of
recurrence that is sufficient to establish a level 2 large deviation principle,
for the absolutely continuous measures. This result encompasses dynamics far
from equilibrium, and thus significantly extends presently known local large
deviations results for quadratic maps.
