Euclidean geometry is needed to describe the trajectory.
First, we determine the intrinsic dimension of the system - the minimum number of degrees of freedom necessary to capture its essential dynamics. We know only how to deal with the transitional regime between regular motions and weak turbulence. We learn how to count and describe itineraries. By now we have covered for the first time the whole distance from diagnosing chaotic dynamics to computing zeta functions. We shall now set up the formalism for evaluating such averages within the framework of the periodic orbit theory.