Chaotic properties of smooth dynamical systems is a very active area of research with many open directions and applications in physics and geometry. This project develops a general framework for studying ergodic and statistical properties of such systems. One of the main motivating questions for the project is: to what extent can a deterministic dynamical system resemble a sequence of independent coin tosses. The main tools for studying this question come from ergodic theory, probability and geometry. The work of the project will result in progress in our understanding of fundamental dynamical phenomena with possible consequences and applications in other areas of mathematics, such as geometry and number theory, and also in physics and economics. The project provides research training opportunities for graduate and undergraduate students and postdoctoral researchers. The project is part of an ongoing program of studying ergodic and statistical properties of smooth systems on manifolds and their interactions with geometry and number theory. The Principal Investigator (PI) focuses on the following three main directions: 1. Chaotic properties for systems with non-zero exponents. Ergodic and statistical properties of smooth systems are quite well understood in the case where all Lyapunov exponents are non-zero (hyperbolic systems). On the other hand, ergodic theory of systems for which some (but not all) exponents are zero is much less understood. The PI studies appearance and