In the last decades uniformly hyperbolic systems have played an important role in the development of the Dynamical Systems Theory. These are systems for which the tangent bundle splits into two invariant sub-bundles, one of them contracting uniformly and the other one expanding uniformly. The study of uniformly hyperbolic systems, its connection with stability and the consequence to the dynamical behavior of systems reached unparallel development. The attention has now shifted beyond uniform hyperbolicity and the persistence of properties that prevail in this wider perspective.
The broadening of the uniformly hyperbolic realm has been carried out by breaking hyperbolicity by means of several mechanisms. One of them consists of combining regions of the phase space where some hyperbolicity remains, together with sets of critical behavior where the uniform hyperbolicity is lost. A powerful way to deal with the lack of hyperbolicity is through probability theory. In fact, the study of observables and their time evolution has become a subject of main interest. Birkhoff’s ergodic theorem establishes the strong law of large numbers in this context, with respect to an invariant probability measure. A case of particular interest in this context, due to its physical meaning, is the study of SRB measures and their properties. Among them we mention: statistical and stochastic stability, decay of correlations, limiting laws, entropy continuity, and multifractal formalism.
Another mechanism of breaking hyperbolicity consists of creation of homoclinic tangencies or heterodimensional cycles, which can generate a rich variety of dynamical phenomena such as intermingled homoclinic classes, robust and persistent heterodimensional cycles, collision and collapse of homoclinic classes, among others [DR1, DR2, DR3]. These are the main mechanisms leading to partially hyperbolic systems: dynamical systems having a dominated splitting of the tangent bundle, combining non-uniform expanding/contracting in a central direction with other of uniform behavior. From a topological viewpoint, a special interest lies in determining if in context of heterodimensional cycles there is some analogous of the well-known Newhouse phenomenon associated to homoclinic tangencies: coexistence of infinitely many attractors with some persistence. Since in this setting there are neither attractors nor repellors, a version of this phenomenon replaces sinks by "elementary pieces of dynamics", where in this case, these pieces are homoclinic classes.
The existence of SRB measures for the class of non-uniformly expanding maps introduced in [V] was obtained in [A1]. In [ABV] it was extended to general classes of non-uniformly expanding maps and partially hyperbolic diffeomorphisms whose central direction is mostly expanding. The method in [ABV] is based on the simple idea of iterating forward the volume measure on some centre-unstable disk. However, it gives not much information on the properties of SRB measures. Remarkable advances in this direction were achieved through the idea of inducing. Roughly speaking, this consists of replacing the initial dynamical system by another one easier to understand and from which one can recover much information on the initial system. This idea goes back to the 70 ́s where Markov partitions were used to study uniformly hyperbolic systems. A main achievement in this direction was obtained in [Y2], where it was developed a framework useful in a systematic treatment of several classes of dynamical systems. A main role in this context is played by Gibbs-Markov structures. Comparing to classical Markov partitions, the big difference lies on the possibility of infinitely many inducing times, as long as the measure of points with larger returns decays to zero. This is fundamental for applications in the non-uniformly hyperbolic context. The results in [Y2] show that if the tail of inducing times decays exponentially fast, then the dynamical system has exponential
rates of mixing. Advances in this direction were obtained in [AP1] for different types of mixing rates.
A main issue in dynamics is the stability of systems. We are particularly interested in the continuous variation of SRB measures, or stationary measures for systems embodying random perturbations. In the uniformly hyperbolic context general results on stochastic and statistical stability were obtained in [Y1]. Results on stochastic stability were drawn in [AA] for non-uniformly expanding maps and in [AAV] for partially hyperbolic diffeomorphisms. Important contributions on the statistical stability of dynamical systems were given in [F,FT] for interval maps, [AV] for multidimensional non-uniformly expanding maps and [ACF] for Hénon attractors. Still in this direction one also mentions [AOT,F] where the entropy of SRB measure is shown to depend continuously on the map for certain families of non-uniformly expanding maps.