"The project deals with the interplay between some local and global aspects of discrete dynamical systems. In the first part we consider several problems related to the Entropy Conjecture. For any map f on a compact manifold M there are some invariants measuring the dynamical complexity at the ‘global’ level (homological or homotopical), like the spectral radius of the map induced on homology, the fundamental-group entropy, etc. In some situations these global invariants give a lower bound for some other ‘local’ invariants, which are in general harder to compute, like the topological entropy or the volume growth. We are interested in studying the relationship between the global and local invariants for different classes of maps, in particular the relationship between the volume growth and the invariants related to the fundamental group of M. On the second part of the proposal we consider partially hyperbolic diffeomorphisms (PHDs). One can associate to the stable and unstable foliations some closed currents or transversal measures, and consequently homology classes. In some situations one can relate the action induced by f on these homology classes with the volume growth of disks inside the leaves of the foliations, and consequently with the topological entropy of the map. For Anosov systems this was done by Shub-Williams and Ruelle-Sullivan, and for PHDs this was done in some cases by Saghin-Xia. We want to investigate when one can establish such relationships, and what are the possible applications of them."