Upper large deviations bound for singular-hyperbolic attracting sets

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 and/or discontinuities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/discontinuous set of the base map. To obtain this upper bound, we show that the base transformation exhibits exponential slow recurrence to the singular set. The results are applied to semiflows modeling singular-hyperbolic attracting sets of $C^2$ vector fields. As corollary of the methods we obtain result on the existence of physical measure for classes of piecewise $C^{1+}$ expanding maps of the interval with singularities and discontinuities. We are also able to obtain exponentially fast escape rates from subsets without full measure.
Comment: 55 pages; 6 figures; accepted version JDDE. Major changes after referee report: Theorem C weakened, new Corollary D included. Deep changes in Section 4 due to issue with bounded distortion estimates. To the best of out knowledge, this is the first time non-exponential initial partitions are used in this setting and still provide upper exponential decay for large deviations