In this seminar we prove the statistical stability of Luzzatto-Viana maps. We show that for each parameter, the map has non-uniform expanding behavior and slow recurrence to the critical/singular set with exponential decay of the measure of the tail set. In this setting, we also obtain transitivity of the map, and so the uniqueness of the physical measure. This improves a result by Araújo, Luzzatto and Viana (2009), where they prove the existence of a finite number of the physical measures. As an application of our results we obtain some statistical properties of the measure via inducing schemes.

References.

[1] Alves, J. (2004). Strong statistical stability of non-uniformly expanding maps. Nonlinearity, 17(4), 1193-1215.

[2] Araújo, V., Luzzatto, S., Viana, M. (2009). Invariant measures for interval maps with critical points and singularities. Advances in Mathematics, 221(5), 1428-1444.

[3] Freitas, J. M. (2005). Continuity of SRB measure and entropy for Benedicks-Carleson quadratic maps. Nonlinearity, 18(2), 831-854.

[4] Luzzatto, S., Viana, M. (2000). Positive Lyapunov exponents-for Lorenz-like families with criticalities. Asterisque, 261(201), 237.