Periodic billiard orbits of self-similar Sierpinski carpets

被引：7

作者：

Chen, Joe P. ^{[1
]}

Niemeyer, Robert G. ^{[2
]}

机构：

[1] Univ Connecticut, Dept Math, Storrs, CT 06269 USA

[2] Univ New Mexico, Dept Math & Stat, Albuquerque, NM 87131 USA

来源：

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | 2014年 / 416卷 / 02期

基金：

美国国家科学基金会;

关键词：

Fractal billiard; Polygonal billiard; Rational (polygonal) billiard; Law of Reflection; Unfolding process; Flat surface; Translation surface; Geodesic flow; Billiard flow; Iterated function system and attractor; Self-similar set; Fractal; Prefractal approximations; Self-similar Sierpinski carpet billiard; Prefractal rational billiard approximations; Sequence of compatible orbits (Eventually) constant sequences of compatible orbits; Footprints; RECURRENCE; FLOWS;

D O I：

10.1016/j.jmaa.2014.03.001

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table Omega(S-a). Based on our refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in S., we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables, Omega(S-a,S-n). The trivial limit of this sequence then constitutes a periodic orbit of Omega(S-a). We also determine the corresponding translation surface S(S-a,S-n) for each prefractal table Omega(S-a,S-n), and show that the genera {gn}(n=0)(infinity)) of a sequence of translation surfaces {S(S-a,S-n)}(n=0)(infinity) increase without bound. Various open questions and possible directions for future research are offered. (c) 2014 Elsevier Inc. All rights reserved.

引用

页码：969 / 994

页数：26

共 21 条

[1] On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces [J].