Large deviations bound for semiflows over a non-uniformly expanding base

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast as time goes to infinity. The arguments need the base transformation to exhibit exponential slow recurrence to the singular set which, in all known examples, implies exponential decay of correlations. Suspension semiflows model the dynamics of flows admitting cross-sections, where the dynamics of the base is given by the Poincaré return map and the roof function is the return time to the cross-section. The results are applicable in particular to semiflows modeling the geometric Lorenz attractors and the Lorenz flow, as well as other semiflows with multidimensional non-uniformly expanding base with non-flat singularities and/or criticalities under slow recurrence rate conditions to this singular/critical set. We are also able to obtain exponentially fast escape rates from subsets without full measure. *The author was partially supported by CNPq-Brazil and FCT-Portugal through CMUP and POCI/MAT/61237/2004.