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A Brief Overview of Outer Billiards on Polygons

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A Brief Overview of Outer Billiards on Polygons Degree project A Brief Overview of Outer Billiards on Polygons Author: Michelle Zunkovic Supervisor: Hans Frisk Examiner: Karl-Olof Lindahl Date: 2015-12-17 Course Code: 2MA11E Subject:Mathematics Level: Bachelor Department Of Mathematics A Brief Overview of Outer Billiards on Polygons Michelle Zunkovic December 17, 2015 Contents 1 Introduction 3 2 Theory 3 2.1 Affine transformations . .3 2.2 Definition of Outer Billiards . .4 2.3 Boundedness and unboundedness of orbits . .4 2.4 Periodic orbits . .6 2.5 The T 2 map and necklace dynamics . .6 2.6 Quasi rational billiards . .8 3 Some different polygons 8 3.1 Orbits of the 2-gon . .8 3.2 Orbits of the triangle . .9 3.3 Orbits of the quadrilaterals . 10 3.3.1 The square . 10 3.3.2 Trapezoids and kites . 10 3.4 The regular pentagon . 11 3.5 The regular septagon . 12 3.6 Structuring all polygons . 13 4 Double kite 13 4.1 Background . 13 4.2 Method . 15 4.3 Result . 15 5 Conclusions 16 2 Abstract Outer billiards were presented by J¨urgenMoser in 1978 as a toy model of the solar system. It is a geometric construction concerning the motions around a convex shaped space. We are going to bring up the basic ideas with many figures and not focus on the proofs. Explanations how different types of orbits behave are given. 1 Introduction It was J¨urgenMoser that aroused the interest of outer billiards to the public. In 1978 Moser published an article named "Is the Solar System Stable?" [1]. He describes the outer billiards as a toy model of the solar system. There are several stability proofs for the solar system but only for limited time. Here the question is what happens with the motion for unlimited time, which is a pure mathematical question and does not necessary have a real world meaning. When studying outer billiards the easiest part is probably to understand the definition. There is still much unknown about the outer billiards and proofs are advanced and will not be the focus in this paper. The basic properties of outer billiards will be discussed and some different examples will be studied as well. In section two the theory will be treated and the definition of an outer billiard, classification of orbits and their motion will be given. In section three special cases of polygons will be brought up and explained, from the easiest case to more complicated ones. In section four we try to find a table that has an unbounded orbit and also try to find this orbit, and at last in section five the conclusions will be given. 2 Theory In this section we are going to define some relevant concepts used to the study of specific polygons in section three. In particular using vector geometry we describe the motion of periodic orbits. 2.1 Affine transformations A transformation A that maps points in the set R2 to itself and whose deter- minant is nonzero is said to be an affine transformation, or an affinity. If X belongs to R2 then A(X) also belongs to R2. With affinity some properties come, one of them is parallelism. That is, if n and m are two parallel lines, then their transformations A(n) and A(m) are parallel lines as well. Another property is the ratio for affine transformations. Given three points on a line, p1; p2 and p3, the ratio between the vectors jp1 − p2j and jp1 − p3j will be the same as the ratio between jA(p1) − A(p2)j and jA(p1) − A(p3)j. In the plane you can displace any given vector in one specific direction by a shear transformation. With a strain transformation you can map the vector 3 (x; y) to (r1x; r2y). A geometric figure can be transformed into a similar figure by a similarity transformation. Any affinity can be described by the product of strain, shear and similarity transformations and for two given triangles 41 and 42, there is an affine transformation from 41 to 42. More about affinities can be found in [2]. A transformation T is a reflection in the origin if it takes a point (x; y) in the plane and map it into (−x; −y). The transformation T commutes with affine transformations A, in the plane. That is, TA = AT , which means that if you apply T first and then the affine transformation A you will get the same result if you first apply the affine transformation A and then T . Affine transformations are important because if two billiards are connected with an affinity then their motions are qualitatively the same. As we will see below the outer billiard transformation is a reflection in a tangency point. If a polygon is a lattice polygon then its vertices lie at rational points, that is the coordinates are rational numbers. When discussing the different polygons in section three we will see that it plays a big role whether the polygon is a lattice polygon or can be transformed into one with an affinity. 2.2 Definition of Outer Billiards The motion around an outer billiard is a reflection through a tangency point on a convex billiard table. The table do not need to be regular or have a special shape. It can be a polygon, ellipse or a combination, for example a half circle. In this paper only polygons will be treated. Choose a starting point x0 outside the table, see figure 1. The point now has two tangency points on the table, and we will choose to go clockwise consistently throughout this article. Reflect x0 in the tangency point γ0, so the distance between x0 and γ0 will be the same as the distance between the new point x1 and γ0. The dynamical system takes x0 to T (x0) = x1. Then x1 will be mapped into x2 in the same way , T stands for tangent map. If a point has more than one tangency point the system is not defined on that point. For a polygon it is the vertices that are the tangency points. In figure 1 the first two iterations are shown. The first and most important thing you want to study is the behavior of the orbits. The question is, what is the character of the different orbits around the table we consider? 2.3 Boundedness and unboundedness of orbits The dynamical system takes a point x ! T (x) ! T 2(x) ! ::: ! T n(x); were T n(x) is the n-fold composition of itself, that is T n = T ◦ T ◦ T ◦ ::: ◦ T : | {z } n times 4 Figure 1: definition of motion. 2 An orbit of a point x0 2 R under T is defined as the set fxngn≥0 where n 0 xn = T (x0) and x0 = T (x0). An orbit can be bounded or unbounded. The latter one is what one could guess, a trajectory that goes to infinity. There are two types of bounded orbits. The periodic ones and the infinite ones. An orbit is said to be periodic with respect to T if xn = x0 for some integer n ≥ 0. It is infinite if you never come back to the same point but the orbit stays bounded in a certain area. A sufficient condition for boundedness is given in section 2:6. With an example one can show that it can occur infinite orbits in a bounded region that never visit the same point twice. One of these is to consider Xn+1 = 2Xn (mod 1) 1 in the binary base where X0 2]0; 1]. For example will 3 = 0:010101::: which we can confirm by rewriting the right hand side to 1 1 1 + + + ::: 4 16 64 1 and by factoring 4 , the sum will be equal to 1 1 1 · 1 = : 4 1 − 4 3 If we compute X1 for X0 = 0:010101:::, the only thing that will happen is the decimal sign will move one step to the right. Since the operation is in mod 1 the integer part will also vanish. Then X1 = 0:101010::: and X2 = 0:010101::: and we see that X0 = X2, hence it is a periodic expansion. Any rational number has a periodic decimal expansion in all bases. Periodic orbits will therefore correspond to the rational numbers and the infinite ones to the irrational numbers. The unique thing with the irrationals is that it is no self repeating in its decimal form. Move the decimal point one step to the right and you will never have the same numbers to the right of the decimal point as you started with. 5 2.4 Periodic orbits When we now have a better understanding of the motion around outer billiards we can give some properties of periodic orbits. Start with a point x0 and an arbitrary outer billiard table. Then x0 should reflect in the corner xc1 and end up in the point x1. This map can be described using vectors. Then the first step, x1 = x0 + 2(xc1 − x0) = 2xc1 − x0; will give us the point x1. We do the same procedure, x2 = 2xc2 − x1 = 2xc2 − 2xc1 + x0; (1) to get the next point. Doing this one more time to then see the pattern, x3 = 2xc3 − x2 = 2xc3 − 2xc2 + 2xc1 − x0: For the n :th point, n+1 n xn = 2(xcn − xcn−1 + ::: + (−1) xc1 ) + (−1) x0: (2) In order for an orbit to be periodic, with period n, x0 = xn. The starting point must be equal to the n:th iteration. If we look at equation (2) we observe that the right hand side will only be equal to x0 when the first parenthesis is equal to zero and n is even.
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