11 papers
A note on the Kronecker--Weyl equidistribution theorem
J. Beck, W. W. L. Chen, Y. Yang
We study the relationship between the discrete and the continuous versions of the Kronecker--Weyl equidistribution theorem, as well as their possible extension to manifolds in high…
Irreversible and dissipative systems
J. Beck, W. W. L. Chen, Y. Yang
We study some new dynamical systems where the corresponding piecewise linear flow is neither time reversible nor measure preserving. We create a dissipative system by starting with…
A note on density of geodesics
J. Beck, W. W. L. Chen, Y. Yang
We extend the famous result of Katok and Zemlyakov on the density of half-infinite geodesics on finite flat rational surfaces to half-infinite geodesics on a finite polycube transl…
Uniformity of geodesic flow in non-integrable 3-manifolds
J. Beck, W. W. L. Chen, Y. Yang
Almost nothing is known concerning the extension of -dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus to non-integrable finite…
Billiards in polyhedra: a method to convert 2-dimensional uniformity to 3-dimensional uniformity
J. Beck, W. W. L. Chen, Y. Yang
The class of 2-dimensional non-integrable flat dynamical systems has a rather extensive literature with many deep results, but the methods developed for this type of problems, both…
Super-fast spreading of billiard orbits in rational polygons and geodesics on translation surfaces
J. Beck, W. W. L. Chen
In this paper, we show that billiard orbits in rational polygons and geodesics on translation surfaces exhibit super-fast spreading, an optimal time-quantitative majority property…
Superdensity and super-micro-uniformity in non-integrable flat systems
J. Beck, W. W. L. Chen
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 in cube-covering systems
J. Beck, W. W. L. Chen
We establish various analogs of the Kronecker-Weyl equidistribution theorem that can be considered higher-dimensional versions of results established in our earlier investigation o…
Time-quantitative density of non-integrable systems
J. Beck, W. W. L. Chen
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 g…
Generalization of a density theorem of Khinchin and diophantine approximation
J. Beck, W. W. L. Chen
The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square exhibits superdensity, which is a best form of time-qua…
Quantitative behavior of non-integrable systems (III)
J. Beck, W. W. L. Chen, Y. Yang
The main purpose of part (III) is to give explicit geodesics and billiard orbits in polysquares that exhibit time-quantitative density. In many instances, we can even establish a b…