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The subject of this monograph is to describe orbits of slowly
chaotic motion. The study of geodesic flow in the unit torus is
motivated by the irrational rotation sequence, where the most
outstanding result is the Kronecker-Weyl equidistribution theorem
and its time-quantitative enhancements, including superuniformity.
Another important result is the Khinchin density theorem on
superdensity, a best possible form of time-quantitative density.
The purpose of this monograph is to extend these classical
time-quantitative results to some non-integrable flat dynamical
systems.The theory of dynamical systems is on the most part about
the qualitative behavior of typical orbits and not about individual
orbits. Thus, our study deviates from, and indeed is in complete
contrast to, what is considered the mainstream research in
dynamical systems. We establish non-trivial results concerning
explicit individual orbits and describe their long-term behavior in
a precise time-quantitative way. Our non-ergodic approach gives
rise to a few new methods. These are based on a combination of
ideas in combinatorics, number theory, geometry and linear
algebra.Approximately half of this monograph is devoted to a
time-quantitative study of two concrete simple non-integrable flat
dynamical systems. The first concerns billiard in the L-shape
region which is equivalent to geodesic flow on the L-surface. The
second concerns geodesic flow on the surface of the unit cube. In
each, we give a complete description of time-quantitative
equidistribution for every geodesic with a quadratic irrational
slope.
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