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We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.
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Consider a finite polysquare or square tiled region, a connected, but not necessarily simply-connected, polygonal region tiled with aligned unit squares. Using ideas from diophantine approximation, we prove that a half-infinite billiard orbit in such a region is superdense, a best possible form of time-quantitative density, if and only if the initial slope of the orbit is a badly approximable number. As the traditional approach to questions of density and uniformity via ergodic theory depends on results such as Birkhoffs ergodic theorem which are essentially time-qualitative in nature and do not appear to lead naturally to time-quantitative statements, we appeal to a non-ergodic approach that is based largely on number theory and combinatorics. In particular, we use the famous 3-distance theorem in diophantine approximation combined with an iterative process. This paper improves on an earlier result of the authors and Yang where it is shown that badly approximable numbers that satisfy a quite severe technical restriction on the digits of their continued fractions lead to superdensity. Here we overcome this technical impediment.
The main purpose of the paper is to give explicit geodesics and billiard orbits in polysquares and polycubes that exhibit time-quantitative density. In many instances of the 2-dimensional case concerning finite polysquares and related systems, we can even establish a best possible form of time-quantitative density called superdensity. In the more complicated 3-dimensional case concerning finite polycubes and related systems, we get very close to this best possible form, missing only by an arbitrarily small margin. We also study infinite flat dynamical systems, both periodic and aperiodic, which include billiards in infinite polysquares and polycubes. In particular, we can prove time-quantitative density even for aperiodic systems.
We show that on any non-integrable finite polysquare translation surface, superdensity, an optimal form of time-quantitative density, leads to an optimal form of time-quantitative uniformity that we call super-micro-uniformity.
This paper is motivated by an interesting problem studied more than 50 years ago by Veech and which can be considered a parity, or mod 2, version of the classical equidistribution problem concerning the irrational rotation sequence. The Veech discrete 2-circle problem can also be visualized as a continuous flat dynamical system, in the form of 1-direction geodesic flow on a 2-square-b surface, a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of barriers and gates on the vertical edges, with appropriate modification of the edge identifications. A famous result of Gutkin and Veech says that 1-direction geodesic flow on any flat finite polysquare translation surface exhibits optimal behavior, in the form of an elegant uniform-periodic dichotomy. However, for irrational values of b, the 2-square-b surface is not a polysquare surface, and Veech and others have highlighted serious violations of the uniform-periodic dichotomy. Here we extend some of the results of Veech to consider cases previously not covered, and also obtain some time-quantitative description of these violations. Furthermore, we establish a far-reaching generalization of some earlier results to the class of flat finite polysquare-b-rational translation surfaces, obtained from flat finite polysquare translation surfaces in a similar way that the 2-square-b surface is constructed.
Arithmetic class are closed subsets of the euclidean space which generalise arithmetical conditions encoutered in dynamical systems, such as diophantine conditions or Bruno type conditions. I prove density estimates for such sets using Dani-Kleinbock-Margulis techniques.
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