• Corpus ID: 219573808

Quantitative behavior of non-integrable systems (III)

@article{Beck2020QuantitativeBO,
  title={Quantitative behavior of non-integrable systems (III)},
  author={Jonathan Beck and W.W.L. Chen and Y. Yang},
  journal={arXiv: Number Theory},
  year={2020}
}
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… 

Figures from this paper

Quantitative behavior of non-integrable systems. II
Here we finish the proof of the Main Theorem on the L-surface, i.e., Theorem 2.1.4 in part (I) of this paper [1], to which the reader is also referred for basic notation. In Sections 3–5 we develop
SUPERDENSITY OF NON-INTEGRABLE BILLIARDS
Consider any rectangular polygon region tiled with unit squares, with possibly some extra walls inside the polygon. We prove that a half-infinite billiard orbit in such a region is superdense, a best
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] exhibits superdensity, which is a best form of time-quantitative density, if
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, .
THE VEECH DISCRETE 2-CIRCLE PROBLEM AND NON-INTEGRABLE FLAT SYSTEMS
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
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 in [1] that a half-infinite geodesic on
The Kronecker-Weyl equidistribution theorem and geodesics in 3-manifolds
Given any rectangular polyhedron 3-manifold P tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these

References

SHOWING 1-10 OF 22 REFERENCES
Quantitative behavior of non-integrable systems. II
Here we finish the proof of the Main Theorem on the L-surface, i.e., Theorem 2.1.4 in part (I) of this paper [1], to which the reader is also referred for basic notation. In Sections 3–5 we develop
Non-ergodic Z-periodic billiards and infinite translation surfaces
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a
Divergent directions in some periodic wind-tree models
The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S.
Dynamics on the infinite staircase
For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite
The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion
Abstract We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely
Introduction to Diophantine Approximation
TLDR
This article formalizes some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals and proves that the inequality |xθ − y| ≤ 1/x has infinitely many solutions by continued fractions.
The invariant measures of some infinite interval exchange maps
We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line
Ergodic infinite group extensions of geodesic flows on translation surfaces
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. K. Frączek and
Ergodicity for infinite periodic translation surfaces
Abstract For a $ \mathbb{Z} $-cover $\widetilde {M} \rightarrow M$ of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the
Topological transitivity of billiards in polygons
Consider a billiard in a polygon Q⊂R2 having all angles commensurate with π. For the majority of initial directions, density of every infinite semitrajectory in configuration space is proved. Also
...
1
2
3
...