Quantitative behavior of non-integrable systems. I

The theory of Uniform Distribution started with the equidistribution of the irrational rotation of the circle, proved around 1905 independently by Bohl, Sierpinski and Weyl. The quantitative upgrading of this qualitative result was done by Hardy, Littlewood, Ostrowski, Weyl, and others around 1920. Their work pointed out the crucial role of continued fractions in uniform distribution (for a good treatment of this, see e.g. the book of Drmota–Tichy [11]). This quantitative theory immediately solves—via “discretization”—the problem of quantitative equidistribution of the torus line flow in a square. Due to a simple equivalence, the torus line also settles the case of all “integrable” flat dynamical systems. In the last forty years a group of (mostly) ergodic theorists worked out a qualitative theory of nonintegrable flat systems (see Section 2.1). Using a new nonergodic approach that we call “shortline method”, here we start to develop a quantitative theory of nonintegrable flat dynamical systems. In this long work—which is a series of several papers—we discuss the similarities and the fundamental quantitative differences between “integrable” and “nonintegrable” systems.