The theory of Dynamical Systems has its origin in the qualitative study of ordinary differential equations or difference equations. Our research addresses these types of systems, with time varying in a continuous, discrete or complex way. We study algebraic, analytic, geometric, probabilistic and topological properties of systems, with some problems arising in Biology, Economics, and Engineering.
A main problem in this field is the existence of physical measures for systems with weak forms of hyperbolicity. Additionally, one aims at deriving some of their features, such as statistical stability, decay of correlations, central limit theorem or large deviations. One is also interested in extreme observations that correspond to the occurrence of rare events.
Another problem that interests us is the transition between different types of dynamics. These may arise through heteroclinic bifurcations, where a connection is created between two or more separated sets. Dynamical systems of special types that are not stable may have their dynamic properties persisting after small changes preserving the type. This happens when the evolution law is Hamiltonian and, more generally, when it preserves area or is reversible. The same holds when the system has symmetry. All these situations are dealt within this line.
Another special type of dynamics addressed is what arises when several low-dimensional dynamical systems (cells) are coupled in a network to create a higher-dimensional system. It may inherit properties both from the individual cells and from the network structure.
The dynamics of holomorphic foliations has a different flavour in that the leaves have real dimension two, so that their fundamental groups are typically 'large' (non-amenable). This fact gives rise to 'large' holonomy groups whose nature of dynamics contrasts with the previous topics. These dynamics tend to have no invariant geometric object and their dynamical and ergodic study follow different lines.