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W. W. L. Chen

Publications and source records attributed to W. W. L. Chen.

12 recordsLinked to original sources
arXiv2024

Billiards in polyhedra: a method to convert 2-dimensional uniformity to 3-dimensional uniformity

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 the traditional approach via Teichmüller geometry and our recent shortline-ancestor method, appear to be exclusively plane-specific. Thus we know very little of any real significance concerning 3-dimensional systems. Our purpose here is to describe some very limited extensions of uniformity in 2 dimensions to uniformity in 3 dimensions. We consider a 3-manifold which is the cartesian product of the regular octagonal surface with the unit torus. This is a restricted system, in the sense that one of the directions is integrable. However, this restriction also allows us to make use of a transference theorem for arithmetic progressions established earlier by Beck, Donders and Yang.

math.DS↗
arXiv2024

A note on the Kronecker--Weyl equidistribution theorem

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 higher dimensions. We also investigate a way to deduce in some limited way uniformity results in higher dimension from results in lower dimension.

math.DS↗
arXiv2024

Uniformity in cube-covering systems

We establish various analogs of the Kronecker-Weyl equidistribution theorem that can be considered higher-dimensional versions of results established in our earlier investigation of the discrete 2-circle problem studied in 1969 by Veech. Whereas the Veech problem can be viewed as one of geodesic flow on a 2-dimensional flat surface, here we study geodesic flow in higher-dimensional flat manifolds. This is more challenging, as the overwhelming majority of the available proof techniques for non-integrable flat systems are based on arguments in dimension 2. For higher dimensions, we need a new approach.

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arXiv2024

Irreversible and dissipative systems

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 finite polysquare translation surface, and then modifying it by including a one-sided barrier on a common vertical edge of two adjacent atomic squares, in the form of a union of finitely many intervals. The line flow in this system partitions the system into a transient set and a recurrent set. We are interested in the geometry of these two sets.

math.DS↗
arXiv2024

Uniformity of geodesic flow in non-integrable 3-manifolds

Almost nothing is known concerning the extension of $3$-dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus $[0,1)^3$ to non-integrable finite polycube translation $3$-manifolds. In the special case when a finite polycube translation $3$-manifold is the cartesian product of a finite polysquare translation surface with the unit torus $[0,1)$, we have developed a splitting method with which we can make some progress. This is a somewhat restricted system, in the sense that one of the directions is integrable. We then combine this with a split-covering argument to extend our results to some other finite polycube translation $3$-manifolds which satisfy a rather special condition and where none of the $3$ directions is integrable.

math.DS↗
arXiv2024

A note on density of geodesics

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 translation $3$-manifold. We also extend this original result to establish a weak uniformity statement.

math.DS↗
arXiv2024

Time-quantitative density of non-integrable systems

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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arXiv2022

Quantitative behavior of non-integrable systems (III)

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 best possible form of time-quantitative density called superdensity. We also study infinite flat dynamical systems, both periodic and aperiodic, which include billiards in infinite polysquare regions. In particular, we can prove time-quantitative density even for aperiodic systems. In terms of optics the billiard case is equivalent to the result that an explicit single ray of light can essentially illuminate a whole infinite polysquare region with reflecting boundary acting as mirrors. In fact, we show that the same initial direction can work for an uncountable family of such infinite systems.

math.NT↗
arXiv2021

Generalization of a density theorem of Khinchin and diophantine approximation

The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square $[0,1]^2$ exhibits superdensity, which is a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. In this paper, we give a proof of the extension of this result of Khinchin to the case when the unit torus $[0,1]^2$ is replaced by a finite polysquare surface, or square tiled surface. The argument is based on diophantine approximation and continued fractions, traditional tools in number theory. In particular, we use the famous $3$-distance theorem in diophantine approximation combined with an iterative process. In short, this is a very number-theoretic study of a very number-theoretic problem. 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 superdense geodesics. Here we overcome this technical impediment. This paper is self-contained, and the reader does not need any knowledge of dynamical systems.

math.DS↗
arXiv2021

New Kronecker-Weyl type equidistribution results and diophantine approximation

An interesting result of Veech more than 50 years ago is a parity, or mod $2$, version of the Kronecker--Weyl equidistribution theorem concerning the irrational rotation sequence $\{qα\}$, $q=0,1,2,3,\ldots.$ If $α$ is badly approximable and $b\in(0,1)$ satisfies $b\ne\{mα\}$ for any $m\in\mathbb{Z}$, then the parity of cardinalities of the sets $\{1\le q\le N:\{qα\}\in[0,b)\}$ as $N\to\infty$ is evenly distributed. We first answer a question of Veech and establish a stronger form of the mod $n$ analog of his result (Theorem 3.1). Furthermore, for irrational $α$ and $b=\{mα\}$ for some $m\in\mathbb{N}$, we give a simple yet precise characterization of those cases that give rise to even distribution (Theorem 2.1). We also obtain time-quantitative description of some very striking violations of uniformity -- this part is particularly number theoretic in nature, and involves Ostrowski representations of positive integers and $α$-expansions of real numbers (Theorem 3.4). The Veech discrete $2$-circle problem can also be visualized as a problem that concerns $1$-direction geodesic flow on a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of symmetric barriers and gates on the vertical edges, with appropriate modification of the vertical edge identifications. We establish a far-reaching generalization of this case to ones that concern $1$-direction geodesic flow on surfaces obtained by modifying a finite square tiled translation surface in analogous but not necessarily symmetric ways (Theorem 3.2).

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