# 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…

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## 7 Citations

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## References

SHOWING 1-10 OF 22 REFERENCES

Quantitative behavior of non-integrable systems. II

- Physics, MathematicsActa Mathematica Hungarica
- 2020

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

- Mathematics
- 2011

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

- Mathematics, Physics
- 2011

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

- Physics
- 2013

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

- Mathematics, Physics
- 2009

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

- Computer Science, MathematicsFormaliz. Math.
- 2015

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

- Mathematics
- 2015

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

- Mathematics
- 2013

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

- MathematicsCompositio Mathematica
- 2013

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

- Mathematics
- 1975

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…