We establish exponential decay of correlations of all orders for locally G-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable distributions of the base Anosov flow are  $C^1$ . This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by infinitesimal transitivity groups used by Dolgopyat, from which we obtain contraction properties for a class of ‘twisted’ symbolic transfer operators.

In this article we study physical measures for $\operatorname {C}^{1+\alpha }$ partially hyperbolic diffeomorphisms with a mostly expanding center. We show that every diffeomorphism with a mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology.

Moreover, for each diffeomorphism with a mostly expanding center, there exists a $C^1$ neighbourhood, such that diffeomorphism among a $C^1$ residual subset of this neighbourhood admits finitely many physical measures, whose basins have full volume.

We also show that the physical measures for diffeomorphisms with a mostly expanding center satisfy exponential decay of correlation for any Hölder observes. In particular, we prove that every $C^2$ , partially hyperbolic, accessible diffeomorphism with 1-dimensional center and nonvanishing center exponent has exponential decay of correlations for Hölder functions.

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

In a general setting of an ergodic dynamical system, we give a more accurate calculation of the speed of the recurrence of a point to itself (or to a fixed point). Precisely, we show that for a certain $\unicode[STIX]{x1D709}$ depending on the dimension of the space, $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,x)=0$ almost everywhere and $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,y)=0$ for almost all $x$ and $y$. This is done by assuming the exponential decay of correlations and making a weak assumption on the invariant measure.

In this paper we prove a Central Limit Theorem forstandard kernel estimates of the invariant density of one-dimensionaldynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergencefor the variance of the estimator and a variation on the Lindeberg–Riomethod. We also give an extension in the case of weaklydependent sequences in a sense introduced by Doukhan and Louhichi.