In the setting of smooth surface diffeomorphisms, we show that the lower semicontinuity defect of the exponents bounds that of the entropy. In particular, Hausdorff dimension is upper semicontinuous among ergodic invariant probability measures with entropy bounded away from zero. We also obtain a new criterion for the existence of an SRB measure with positive entropy. Joint work with Sylvain CROVISIER and Omri SARIG.