Spectral estimates for Ruelle operators with two parameters and sharp large deviations

We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + ıb)\tau + (c + ıd) g}$ with two complex parameters and Hölder functions $f,\: g$ generalizing the case $\Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].