Quantitative behavior of non-integrable systems. II

  title={Quantitative behavior of non-integrable systems. II},
  author={Jonathan Beck and Merijn E. Donders and Y. Yang},
  journal={Acta Mathematica Hungarica},
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 the basic form of the shortline method. In the subsequent papers this basic form will be be extended and modified in many different ways. 

Figures from this paper

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
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
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
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
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, .
Uniformity of 3-dimensional billiards
Given any polycube, i.e., a rectangular polyhedron tiled with unit cubes, we exhibit infinitely many explicit directions such that every half-infinite billiard orbit with any such initial direction
Quantitative behavior of non-integrable systems (III)
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


Quantitative behavior of non-integrable systems. I
The theory of Uniform Distribution started with the equidistribution of the irrational rotation of the circle, proved around 1905 independently by Bohl, Sierpinski and Weyl. The quantitative
Cubic curves and totally geodesic subvarieties of moduli space
In this paper we present the rst example of a primitive, totally geodesic subvariety F M g;n with dim(F ) > 1. The variety we consider is a surfaceF M 1;3 dened using the projective geometry of
Minimal non-ergodic directions on genus-2 translation surfaces
It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for any genus-2 translation surface which is not a Veech surface
Billiards on almost integrable polyhedral surfaces
The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We
Weak mixing directions in non-arithmetic Veech surfaces
We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon).
Billiards and Teichmüller curves on Hilbert modular surfaces
This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmuller curves lie on Hilbert modular surfaces
A condition for unique ergodicity of minimal symbolic flows
A sufficient condition for unique ergodicity of symbolic flows is provided. In an important but special case of interval exchange transformations, the condition has already been validated by W. Veech.
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
There exists a Teichmuller discΔ n containing the Riemann surface ofy2+x n =1, in the genus [n−1/2] Teichmuller space, such that the stabilizer ofΔ n in the mapping class group has a fundamental
Veech surfaces and complete periodicity in genus two
We announce a classification of genus 2 Veech surfaces in the stratum with a single double zero. Furthermore, we classify all completely periodic translation surfaces in genus 2.
Prym Varieties and Teichmüller Curves
This paper gives a uniform construction of infinitely many primitive Teichmuller curves V ⊂ Mg for g = 2, 3 and 4.