Abstract: In dealing with nonlinear dynamics people are used to its complexity and ask questions only on their asymptotic in time properties. Answers to these questions occur to be still very difficult to obtain. In fact, even most of notions used in nonlinear dynamics (like Lyapunov exponents, decay of correlations, etc) already involve an infinite time limit. I'll demonstrate that one could be more ambitious and some sensible questions on a finite time dynamics could be fairly completely answered. In fact, we even demonstrate new results on finite time evolution of Markov chains, which always were considered as simple approximations of chaotic dynamical systems. A major motivation for this new area of research came from the attempts to dynamically (rather than statically) characterize nodes and edges of dynamical networks.