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# Outer Billiards on Kites (AM-171)

Richard Evan Schwartz
Copyright Date: 2009
Pages: 312
https://www.jstor.org/stable/j.ctt7t77g

## Table of Contents

1. Front Matter
(pp. i-iv)
2. Table of Contents
(pp. v-x)
3. Preface
(pp. xi-xiv)
4. Chapter One Introduction
(pp. 1-16)

B. H. Neumann [N] introducedouter billiardsin the late 1950s. In the 1970s, J. Moser [M1] popularized outer billiards as a toy model for celestial mechanics. See [T1], [T3], and [DT1] for expositions of outer billiards and many references on the subject.

Outer billiards is a dynamical system defined (typically) in the Euclidean plane. Unlike the more familiar variant, which is simply calledbilliards, outer billiards involves a discrete sequence of moves outside a convex shape rather than inside it. To define an outer billiards system, one starts with a bounded convex set$K\; \subset {{\bf{R}}^2}$and considers a point${x_0} \in {{\bf{R}}^2} - K$....

5. ### PART 1. THE ERRATIC ORBITS THEOREM

• Chapter Two The Arithmetic Graph
(pp. 19-32)

LetPbe a convex polygon. We denote the outer billiards map relative toPby$\psi '$, and the square of the outer billiards map by$\psi = {(\psi ')^2}$. Our convention is that a person walking frompto$\psi '(p)$sees thePon the right side. These maps are defined away from a countable set of line segments in${{\bf{R}}^2} - P$. This countable set of line segments is sometimes called thelimit set.

The result in [VS], [K], and [GS] states, in particular, that the orbits for rational polygons are all periodic. In this case, the complement of the limit set is...

• Chapter Three The Hexagrid Theorem
(pp. 33-40)

In this section we describe a certain quadrilateral, which we call thearithmetic kite. This object is meant to “live” in the same plane as the arithmetic graph. The diagonals and sides of this quadrilateral define 6 special directions. In the next section we describe a grid made from 6 infinite families of parallel lines based on these 6 directions. Let$A={p {\left/\right.}q}$. Figure 3.1 accurately shows$\mathcal{K}{\text{(A)}}$forA= 1/3.

One can construct this figure using straight lines and the given coordinates. The pairs of lines that look parallel are supposed to be parallel. Setting$a = (q,q),$we have

$b = \frac{{a\; - \;V}}{2},\quad \quad U = Aa\; + \;(1 - A)b,\quad \quad W = \frac{U}{{1 + A}} = \frac{{b + c}}{2}$....

• Chapter Four Period Copying
(pp. 41-44)

We discussed inferior and superior sequences in §1.4. Here we say a bit more. Let$p/q \in (0,\;1)$be any odd rational. There are unique rationals${p_ - }/{q_ - }$and${p_ + }/{q_ + }$such that

$\frac{{{p_ - }}} {{{q_ - }}} < \frac{p} {q} < \frac{{{p_ + }}} {{{q_ + }}},\quad \quad {\text{max(}}{q_ - },\;{q_ + }) < q,\quad \quad q{p_ \pm } - p{q_ \pm } = \pm 1$. (4.1)

See Chapter 17 for more details.

We define the odd rational.

$\frac{{p'}} {{q'}} = \frac{{|{\kern 1pt} {p_ + } - {p_ - }{\kern 1pt} |}} {{|{\kern 1pt} {q_ + } - {q_ - }{\kern 1pt} |}}$, (4.2)

where$p'/q'$is the unique odd rational satisfying the equation

$q' < q,\quad \quad |{\kern 1pt} pq' - qp'{\kern 1pt} |\; = 2$. (4.3)

We call$p'/q'$theinferior predecessorof$p/q$, and we write$p'/q' \leftarrow p/q$or$p/q \to p'/q'$. We can iterate this procedure. Any$p/q$belongs to a finite chain

$\frac{1} {1}\; \leftarrow \;\frac{{{p_1}}} {{{q_1}}} \leftarrow \cdots \leftarrow \frac{{{p_n}}} {{{q_n}}} = \frac{p} {q}$. (4.4)

Corresponding to this sequence, we define

${d_k} = {\text{floor}}\left( {\frac{{{q_{k + 1}}}} {{2{q_k}}}} \right)$. (4.5)

We define thesuperior predecessorof$p/q$to...

• Chapter Five Proof of the Erratic Orbits Theorem
(pp. 45-52)

In this section we will prove the following result.

Lemma 5.1Suppose that A is the limit of a strong sequence$\{ {A_n}\}$.Then statement 1 of the Erratic Orbits Theorem holds for A.

Statement 1 of the Erratic Orbits Theorem follows from Theorem 4.2 and Lemma 5.1. The reader who is satisfied with the Erratic Orbits Theorem for all$A \in {\Delta _2}$can use the much easier Lemma 4.3 in place of Theorem 4.2.

In our proof of Lemma 5.1, we will consider the monotone increasing case. The monotone decreasing case is essentially the same. Let$\varepsilon = 1/8$be as in the definition...

6. ### PART 2. THE MASTER PICTURE THEOREM

• Chapter Six The Master Picture Theorem
(pp. 55-68)

Recall that$\Xi \; = \;{{\mathbf{R}}_ + }\;X\;\{ - 1,\;1\}$. We distinguish two special subsets of$\Xi$.

${\Xi _ + } = \bigcup\limits_{k = 0}^\infty {(2k,\:2k + 2) \times \{ {{( - 1)}^k}\} }$,${\Xi _ - } = \bigcup\limits_{k = 1}^\infty {(2k,\:2k + 2) \times \{ {{( - 1)}^{k - 1}}\} }$. (6.1)

Each set is an infinite disconnected union of open intervals of length 2. The reflection in thex-axis interchanges${\Xi _ + }$and${\Xi _ - }$. The union${\Xi _ + }\; \cup \;{\Xi _ - }$partitions the set$({{\mathbf{R}}_ + } - 2{\mathbf{Z}}{\text{)}}\: \times \:\{ \pm 1\}$.

Define

${R_A} = [0,\;1\; + \;A]\; \times \;[0,\;1\; + \;A]\; \times \;[0,\;1]$. (6.2)

${R_A}$is a fundamental domain for the action of a certain lattice${\Lambda _A}$. This lattice is defined by the following matrix.

${\Lambda _A} = \left[ {\begin{array}{*{20}{c}} {1 + A} & {1 - A} & { - 1} \\ 0 & {1 + A} & { - 1} \\ 0 & 0 & 1 \\ \end{array} } \right]\,{{\mathbf{Z}}^3}.$. (6.3)

We mean to say that${\Lambda _A}$is the Z-span of the column vectors of the above matrix.

We define maps

${\mu _ \pm }\,:\;{\Xi _ \pm } \to {R_A}$(6.4)

by the equations

${\mu _ \pm }(t,\;*) = \left( {\frac{{t - 1}} {2},\;\frac{{t + 1}} {2},\;\frac{t} {2}} \right)\; \pm \;\left( {\frac{1} {2},\;\frac{1} {2},\;0} \right)\quad {\text{mod}}\;\Lambda$. (6.5)

The maps depend on only...

• Chapter Seven The Pinwheel Lemma
(pp. 69-76)

The Pinwheel Lemma gives a formula for the return map$\Psi \,:\Xi \to \Xi$in terms of maps we callstrip maps. Similar objects are considered in [GS] and [S].

Consider a pair$(\Sigma ,\;L)$, where$\Sigma$is an infinite planar strip andLis a line transverse to . The pair$(L,\;\Sigma )$determines two vectors${V_ + }$and${V_ - }$, each of which points from one boundary component of$\Sigma$to the other and is parallel toL. Clearly,${V_ - } = - {V_ + }$.

See Figure 7.1.

For almost every point$p \in {{\bf{R}}^2}$, there is a integernsuch that

$E(p):\, = p + n{V_ + } \in \Sigma$. (7.1)

We callEthe strip map defined relative...

• Chapter Eight The Torus Lemma
(pp. 77-84)

For ease of exposition, we state and prove the (+) halves of our results. The (-) halves have the same formulation and proof.

Let${\mu _ + }$be as in Equation 6.5. We write${({\mu _ + })_A}$to emphasize the dependence on the parameterA. Let${T^4} = \hat R/\Lambda$, the 4-dimensional quotient discussed in §6.7. Topologically,${T^4}$is the product of a 3-torus with (0, 1 ). We now define

${\mu _ + }:{\Xi _ + } \times (0, 1) \to {T^4}$

by the obvious formula

${\mu _ + }(p,\;A)\; = \;({({\mu _ + })_A}(p),\;A)$. (8.1)

We are just stacking all these maps together.

Referring to the Pinwheel Lemma, we have$\Psi (p)\; = \;\chi \circ {E_8} \ldots {E_1}(p)$whenever both maps are defined. Let$p \in {\Xi _ + }$. We set${p_0} = p$and inductively define...

• Chapter Nine The Strip Functions
(pp. 85-92)

The purpose of this chapter is to understand the functions${\sigma _j}$that arose in the proof of the Torus Lemma. See Equation 8.21. We continue using the notation from the previous chapter. We call these functionsstrip functions. Let$\left\langle x \right\rangle$denote the fractional part ofx. Sometimes we interpret$\left\langle x \right\rangle$as an element of${\text{R/Z}}$.

Let${W_k} \subset \;{\Xi _ + }\; \times \;(0,\;1)$denote the set of points where${E_k} \ldots {E_1}$is defined but${E_{k + 1}}{E_k} \ldots {E_1}$is not defined. Let${S_k}$denote the closure of${\mu _ + }({W_k})$inR. HereRis as in Equation 6.6. Finally, let

${{W'}_k} = \bigcup\limits_{j = 0}^{k - 1} {{W_j},} \quad \quad {{S'}_k} = \bigcup\limits_{j = 0}^{k - 1} {{S_j},\quad \quad k = 1,\; \ldots ,\;7}$. (9.1)

The Torus Lemma applies to any point...

• Chapter Ten Proof of the Master Picture Theorem
(pp. 93-98)

First we recall some notation from previous chapters.

LetSbe the singular set defined in Equation 9.2.

Let$\hat S$denote the union of hyperplanes listed in Chapter 6.2.

Letddenote distance on the polytopeR.

Let$\theta (p,\;A)$be the quantity from the Torus Lemma in §8.

Below we will establish the following result.

Lemma 10.1 (Hyperplane)$S \subset \;\hat S$and$\theta (p,\;A) \ge \;d({\mu _ + }(p,\;A),\;\hat S)$.

The Hyperplane Lemma essentially says that the singular set is small and simple. Before we prove the Hyperplane Lemma, we will finish the proof of the Master Picture Theorem.

Say that aball of constancyin$R\; - \;\hat S$is an...

7. ### PART 3. ARITHMETIC GRAPH STRUCTURE THEOREMS

• Chapter Eleven Proof of the Embedding Theorem
(pp. 101-106)

Let$\hat \Gamma \; = \;{\hat \Gamma _\alpha }(A)$be the arithmetic graph for a parameterAand some$\alpha \notin 2{\bf{Z}}[A]$. The reader will see from our proof that the choice of$\alpha$is not important. As a first step in the proof of the Embedding Theorem, we show that all nontrivial vertices of${\hat \Gamma }$have valence 2. Dynamically, a vertex of valence 1 corresponds to a point$x \in \Xi$such that$x \ne \Psi (x)\; = \;{\Psi ^{ - 1}}(x)$.

Let$p \in {{\bf{Z}}^2}$be a nontrivial vertex of${\hat \Gamma }$. Let${q_ + }$and${q_ - }$be the two neighbors ofp. We would like to show that${\hat \Gamma }$has valence 2 atp. If this fails, then we must...

• Chapter Twelve Extension and Symmetry
(pp. 107-116)

Referring to §6.6, the maps${M_ + }$and${M_ - }$are defined on all of${{\bf{Z}}^2}$. This gives an extension of the arithmetic graph to all of${{\bf{Z}}^2}$. We denote this full extension by${\hat \Gamma }$.

Figure 12.1 shows$\hat \Gamma (3/7)$, as well as the hexagrid$G(3/7)$, from §3.1. The bottom of the shaded parallelogram is the baseline. In the rational case, both the arithmetic graph and the hexagrid are invariant under a certain lattice$\Theta$of translations of${{\bf{Z}}^2}$. The shaded parallelogram is the fundamental domain for$\Theta$. In this section we give the formulas foe the lattice and establish the translational symmetry....

• Chapter Thirteen Proof of Hexagrid Theorem I
(pp. 117-124)

The proof of Hexagrid Theorem I is the same in the odd and even cases.

Say that afloor lineis a negatively sloped line of the floor grid. Floor lines all have slope$- A$. Say that afloor pointis a point on a floor line. Such a point need not have integer coordinates. The maps${M_ + }$and${M_ - }$from §6.6 are constant on floor lines. Thus, ifLis a floor line,${M_ \pm }(L)$is a single point.

Lemma 13.1If p isa floor point, then${M_ - }(p)\; \equiv \;(\beta ,\;0,\;0)$mod$\Lambda$for some$\beta \in \;{\bf{R}}$

Proof: Suppose first that$p/q$is odd....

• Chapter Fourteen The Barrier Theorem
(pp. 125-132)

Let p/q be an even rational. Let$V\; = \;(q,\; - p)$. Referring to Equation 4.1, one of the two rationals p±/q± is even and one is odd. Let$p'/q'$denote whichever of these rationals is odd. (We allow the case when$p'/q' = 1/1$.) We call$p'/q'$theodd predecessorof$p/q$. We say that the barrier is the line parallel toVthat contains the point

$\left( {0,\;\frac{{p'\; + \;q'}} {2}} \right)$. (14.1)

Theorem 14.1 (Barrier)Let e be an edge of$\hat \Gamma (p/q)$that crosses the barrier line. Then there is some$k \in \;{\mathbf{Z}}$such that the translate$e\; + \;kV$is an edge of$\Gamma (p/q)$Moreover, there are only two such...

• Chapter Fifteen Proof of Hexagrid Theorem II
(pp. 133-142)

We will prove statement 2 of the Hexagrid Theorem for odd rationals. The even case has an essentially identical proof. Here we remark on one small difference. Call a point in R²badif it has the form(m,y), whereyis a half-integer. According to statement 3 of Lemma 15.1 below, a door cannot be a bad point in the odd case. In the even case, we simply declare that a door cannot be a bad point. See the definition in Chapter 3. Having ruled out the bad points in both cases, our proof is practically independent of parity....

• Chapter Sixteen Proof of the Intersection Lemma
(pp. 143-150)

One way to prove the Intersection Lemma is just by inspection. One can play with Billiard King and see that the result is true. Given the simple nature of the partitions involved, a falsehood in the lemma would be easily detectable by a small amount of experimentation.

Rather than just appeal to experimentation, we will explain a proof that involves finding the intersection patterns of finitely many convex lattice polytopes in R⁴. The proof we give is similar to that presented in previous chapters. Previously, e.g., in §11.1, our method was straightforward. Here there is a technical complication that we...

8. ### PART 4. PERIOD-COPYING THEOREMS

• Chapter Seventeen Diophantine Approximation
(pp. 153-162)

We will describe a hyperbolic geometry construction of the inferior sequence defined in §4.1. Our proof is similar to that for ordinary continued fractions. See [BKS]. Also, see [Be] for background on hyperbolic geometry, and [Da] for the classic theory of continued fractions.

Our model for the hyperbolic plane is the upper half-plane${{\bf{H}}^2}\; \subset \;{\bf{C}}$. The group$S{L_2}({\bf{R}})$acts isometrically by linear fractional transformations. The geodesics are vertical rays or semicircles centered on R. TheFarey graphis a tiling of${{\bf{H}}^2}$by ideal triangles. We join${p_1}/{q_1}$and${p_2}/{q_2}$by a geodesic iff${p_1}{q_2} - {p_2}{q_1}{\kern 1pt} |\; = \;1$. The resulting graph divides the hyperbolic...

• Chapter Eighteen The Diophantine Lemma
(pp. 163-170)

Let$p/q$be an odd rational.

Consider the following linear functionals.

$F(m,\;n)\; = \;\left( {\frac{p}{q},\;1} \right)\; \cdot \;(m,\;n)$. (18.1)

$G(m,\;n)\; = \;\left( {\frac{{q\; - \;p}}{{p\; + \;q}},\;\frac{{ - 2q}}{{p\; + \;q}}} \right)\; \cdot \;(m,\;n)$. (18.2)

$H(m,\;n)\; = \;\left( {\frac{{ - {p^2} + \;4pq\; + \;{q^2}}}{{{{(p\; + \;q)}^2}}},\;\frac{{2q(q\; - \;p)}}{{{{(p\; + \;q)}^2}}}} \right)\; \cdot \;(m,\;n)$. (18.3)

We have$F\; = \;(1/2)M$, whereMis the fundamental map from Equation 2.10.

We can understandGandHby evaluating them on a basis.

$H(V)\; = \;G(V)\; = \;q;\quad \quad H(W)\; = - G(W)\; = \;\frac{{{q^2}}}{{p\; + \;q}}$. (18.4)

Here$V\; = \;(q,\; - {\kern 1pt} p)$andWare the vectors from Equation 3.2. We can also understandGby evaluating on a simpler basis.

$G(q,\; - {\kern 1pt} p)\; = \;q;\quad \quad \;G( - 1,\; - 1)\; = \;1$. (18.5)

We can also (further) relateGandHto the haxagrid in Chapter 3. A direct calculation establishes the following result.

Lemma 18.1The fibers of G are parallel to the top left edge...

• Chapter Nineteen The Decomposition Theorem
(pp. 171-180)

The Room Lemma confines one period of$\Gamma (p/q)$to a certain parallelogram$R(p/q)$when$(p/q)$is odd. In this section we explain a sharper result, along the same lines, that confines one period of$\Gamma (p/q)$to a union of two parallelograms. The reader might want to glance at Figure 19.1 before reading the definitions that follow.

Given an odd rational$A = (p/q)$, we construct the even rationals${A_ \pm } = {p_ \pm }/{q_ \pm }$. We let${A'}$be the inferior predecessor ofA, and we let$A{\kern 1pt} *$be the superior predecessor, as in §4.1. For each rational, we use Equation 3.2 to construct the correspondingVand...

• Chapter Twenty Existence of Strong Sequences
(pp. 181-184)

In this chapter, we prove Theorem 4.2. For the sake of efficiency, our proof will be essentially algebraic. However, a clear geometric picture underlies our constructions. We discussed this geometric picture in §19.2. The reader might want to reread that section before going through the proof here. Also, the reader might want to review the proof we gave of Lemma 4.3 in § 18.3. Our proof here is similar to the one given there.

LetAbe any irrational parameter. Let$\{ {p_n}/{q_n}\}$denote the superior sequence associated toA. LetSbe a monotone subsequence of the superior sequence. We...

9. ### PART 5. THE COMET THEOREM

• Chapter Twenty-One Structure of the Inferior and Superior Sequences
(pp. 187-192)

Let$\{ {p_n}/{q_n}\}$be the inferior sequence associated to an irrational parameter$A\; \in \;(0,\;1)$and let$\{ {d_n}\}$be the sequence obtained from Equation 4.5. We call$\{ {d_n}\}$the inferior renormalization sequence. We call the subsequence of$\{ {d_n}\}$corresponding to the superior terms thesuperior renormalization sequenceor just therenormalization sequence. Referring to the inferior sequences, we have${d_n}\; = \;0$if and only ifnis not a superior term. In this case, we callnaninferior term. So, the renormalization sequence is created from the inferior renormalization sequence simply by deleting all the 0s.

For any odd rational$p/q \in \;(0,\;1)$, define

$p{\kern 1pt} *\; = \;{\rm{min(}}{p_ - },\;{p_ + });\quad \quad q{\kern 1pt} * = {\rm{min(}}{q_ - },\;{q_ + })$....

• Chapter Twenty-Two The Fundamental Orbit
(pp. 193-204)

We will assume that$p/q = {p_n}/{q_n}$, thenth term in a superior sequence. We call${O_2}(1/{q_n}, - 1)$thefundamental orbit. Let${C_n}$denote the set from Theorem 1.8. Let

${{C'}_n} = {O_1}(1/{q_n}, - 1) \cap I,\quad \quad I = [0,\;2]\; \times \;\{ - 1\}$. (22.1)

Theorem 1.8 says that${C_n} = {{C'}_n}$. In this chapter, we will prove Theorem 1.8 and establish some geometric results about how the orbits return to${C_n}$.

After we prove Theorem 1.8, we will establish a coarse model for how the points of${O_2}(1/{q_n})$return to${C_n}$. Statement 2 of the Comet Theorem is s kind of geometric limit of the Discrete Theorem, and statement 3 of the Comet Theorem is the “geometric...

• Chapter Twenty-Three The Comet Theorem
(pp. 205-218)

We fix an irrational parameter$A \in \;(0,\;1)$. Let$\{ {A_n}\}$be the superior sequence approximatingA. Let${{\hat \Gamma }_n}$be the arithmetic graph corresponding to${A_n}$. We say that a vertex$\upsilon$of${{\hat \Gamma }_n}$isD-low if$\upsilon$is withinDvertical units of the baseline of${{\hat \Gamma }_n}$.

Noe that the low vertices considered in the previous chapter are 1-low vertices. These vertices play a special role in our arguments. The fundamental mapMfrom Equation 2.10 maps the 1-low vertices into the interval

$J\; = \;(0,\;2)\; \times \;\{ - 1,\;1\}$. (23.1)

In Part 6 we prove the following result.

Theorem 23.1 (Low Vertex)Fix${N_0}$.There are constants...

• Chapter Twenty-Four Dynamical Consequences
(pp. 219-226)

Here we prove Theorem 1.3. Statement 3 of this Theorem is contained in the Comet Theorem. We just have to prove statements 1 and 2.

Recall from the introduction that a set$S \subset \;{{\bf{R}}^2}$islocally homogeneousif every two points ofShave arbitrarily small neighborhoods that are translation equivalent. Note that the points themselves need not be in the same positions within these sets.

Statements 1 and 2 of Theorem 1.3 say, respectively, that${U_A}$is dynamically minimal and locally homogeneous. Statement 3 of Theorem 1.3 is an immediate consequence of the Comet Theorem.

Proof of Statement 1: Since...

• Chapter Twenty-Five Geometric Consequences
(pp. 227-236)

Here we prove statement 1 of Theorem 1.5.

Lemma 25.1${U_A}$has length 0.

Proof: Since${U_A}$is locally homogeneous, it suffices to prove that${C_A}$has length 0. Let${\lambda _n}\; = \;|{\kern 1pt} A{q_n}\; - \;{p_n}{\kern 1pt} |$, as in Equation 21.5. We define

${G_n}\; = \;\sum\limits_{k = n + 1}^\infty {2{\lambda _k}{d_k}}$. (25.1)

Then

${C_A} \subset \;\sum\limits_{\kappa \in {\Pi _n}} {\left( {{I_n}\; + \;X(\kappa )} \right)}$. (25.2)

Here${I_n}$is the interval with endpoints (0, 1) and$({G_n},\;1)$. In other words,${C_A}$is contained in${D_n}$translates of an interval of length${G_n}$. We just need to prove that${D_n}{G_n} \to 0$. It suffices to prove this whennis even. By Equation 21.6,

${D_n}\; < \;{\varepsilon ^{ - n}}{q_n},\quad \quad \varepsilon \; = \;\sqrt {5/4}$. (25.3)

By Equation 21.5 we have

${G_n}\; < \;2\;\sum\limits_{k = n + 1}^\infty {q_k^{ - 1}} < \;2q_n^{ - 1}\sum\limits_{k = 1}^\infty {{2^{ - k}}} < \;2q_n^{ - 1}$. (25.4)

Here we have used...

10. ### PART 6. MORE STRUCTURE THEOREMS

• Chapter Twenty-Six Proof of the Copy Theorem
(pp. 239-248)

LetAbe an odd rational. Let${A_ - }$be as in Equation 4.1. Let${V_ - } = ({q_ - },\; - {p_ - })$. Here we give a formula for the pivot points${E^ \pm }$associated toA. Recall that these points are the endpoints of the pivot arc, the subject of the Copy Theorem.

Lemma 26.1The following are true.

If${q_ - } < \;{q_ + }$,then${E^ + } + \;{E^ - } = - {V_ - }\; + \;(0,\;1)$.

If${q_ + } < \;{q_ - }$,then${E^ + } + \;{E^ - } = {V_ + }\; + \;(0,\;1)$.

Proof: We will establish this result inductively. Suppose first that${\rm{1/1}} \leftarrow {\rm A}$. Then

$A = \frac{{2k\; - \;1}}{{2k\; + \;1}},\quad \;{E^ - } = ( - k,\;k),\quad \quad {E^ + } = (0,\;0),\quad \quad {V_ - } = (k,\; - k\; + \;1)$.

${A_ - } = \frac{{k\; - \;1}}{k},\quad \quad {q_ - } = k\; - \;1\; < \;k\; = \;{q_ + }$.

The result works in this case.

In general, we have

$A = {A_2},\quad \quad \quad {A_0} \leftarrow \;{A_1} \leftarrow {A_2}$.

There are 4 cases, depending on Lemma 17.2. Here the index is$m\; = \;1$. We...

• Chapter Twenty-Seven Pivot Arcs in the Even Case
(pp. 249-258)

Given two rationals${A_1} = {p_1}/{q_1}$and${A_2}\; = \;{p_2}/{q_2}$, we introduce the notation

${A_1}\, \vdash \,{A_2}\quad \quad \Leftrightarrow \quad \quad |{\kern 1pt} {p_1}{q_2} - {q_1}{p_2}{\kern 1pt} |\; = \;1,\quad \;{q_1} < {q_2}$. (27.1)

In this case, we say that${A_1}$and${A_2}$areFarey-related. We sometimes call$({A_1},\;{A_2})$aFarey pair.

We have the notions ofFarey additionandFarey subtraction, respectively.

${A_1} \oplus \;{A_2}\; = \;\frac{{{p_1}\; + \;{p_2}}}{{{q_1}\; + \;{q_2}}},\quad \quad {A_2} \ominus {A_1}\; = \;\frac{{{p_2} - {p_1}}}{{{q_2} - {q_1}}}$. (27.2)

Note that${A_1}\, \vdash \,{A_2}$implies that${A_1} \vdash ({A_1} \oplus \;{A_2})$and that${A_1}$is Farey-related to${A_2} \ominus {A_1}$.

Lemma 27.1Let${A_1}$be an even rational. Then there is a unique odd rational${A_2}$such that${A_1}\, \vdash \,{A_2}$and$2{q_1} > {q_2}$. In this case, we write${A_1} \models {A_2}$.

Proof: Equation 4.1 works for both even and odd rationals. When${A_1}$is even, exactly one of...

• Chapter Twenty-Eight Proof of the Pivot Theorem
(pp. 259-272)

We first prove the Pivot Theorem for$A\; = \;1/q$. This case does not fit the pattern we discuss below.

Let$\Gamma$be the arithmetic graph associated to$A\; = \;1/q$and let$P\Gamma$denote the pivot arc. In all cases,$P\Gamma$contains the vertices (0, 0) and (-1, 1). These vertices correspond to the two points

$\left( {\frac{1}{q},\; - 1} \right),\quad \quad \left( {2 - \frac{1}{q},\; - 1} \right)$. (28.1)

These two points are the midpoints of the special intervals

${I_1} = \left( {0,\;\frac{2}{q}} \right)\; \times \;\{ - 1\} ,\quad \quad \;{I_2} = \left( {2 - \frac{2}{q},\;2} \right)\; \times \;\{ - 1\}$. (28.2)

Byspecial interval, we mean intervals in the sense of §2.2. Recall from that section that these special intervals are permuted by the outer billiards map.

The special intervals in Equation 28.2 appear...

• Chapter Twenty-Nine Proof of the Period Theorem
(pp. 273-278)

LetAbe some rational parameter. For each polygonal low component$\beta$of$\Gamma (A)$, we define the pivot arc$P\beta$to be the lower arc of$\beta$that joins the two low vertices that are farthest apart. We saylower arcbecause all the components are closed polygons, and hence two arcs join the pivot points in all cases. WhenAis an even rational and$\beta = \Gamma$, this definition coincides with the definition of$P\Gamma$, by the Pivot Theorem. In general, we say that a pivot arc of$\Gamma$is a pivot arc of some low component of${\hat \Gamma }$. We...

• Chapter Thirty Hovering Components
(pp. 279-286)

Let$A \in \;(0,\;1)$be a rational parameter. We say that$\upsilon \in \;{{\bf{Z}}^2}$isD-lowif the baseline of$\Gamma (A)$separates$\upsilon$from$\upsilon \; - \;(0,\;D)$. Here$D \in \;{\bf{Z}}$. We have the usual convention that the baseline is the line of slope -Athrough the point$(0,\; - {\kern 1pt} \varepsilon )$, where$\varepsilon$is an infinitesimally small positive number. Thus (0,0) is 1-low. Previously, we were interested in 1-low vertices, which we calledlow.

Let$\beta$be a component of$\hat \Gamma (A)$. We call$\beta$a hovering component if it has no 1-low vertices. More specifically, we call$\beta$aD-hovering componentof$\hat \Gamma (A)$if$\beta$has no 1-low vertices and...

• Chapter Thirty-One Proof of the Low Vertex Theorem
(pp. 287-294)

The Low Vertex Theorem in Chapter 23 is a consequence of the following result.

Lemma 31.1 (Descent)Let$A \in \left( {0,1} \right)$be irrational. Let$\left\{ {{B_n}} \right\}$be any sequence of rationals in (0, 1) that converges toA. Letβbe a low component of$\hat \Gamma ({B_n})$There is some constant$D'$such that everyD-low vertex ofβin less than$D'$steps. Here$D'$depends onDand onAbut not onn.

Proof of the Low Vertex Theorem: Let${N_0}$and$\{ {\upsilon _n}\}$be as in the Low Vertex Theorem. Let${\beta _n}$be the component of${\hat \Gamma _n}$that contains${\upsilon _n}$....

11. Appendix
(pp. 295-302)
12. Bibliography
(pp. 303-304)
13. Index
(pp. 305-306)