By K. Ueno

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**Example text**

Statement 1 of the Hexagrid Theorem says that the edges of incident to ζk lie between L k and L k+1 (rather than between L k−1 and L k ). We compute that M1 (ζk ) = k( p + q). 9. 5, statement 1 of the Hexagrid Theorem, and our remarks about ζk . 9 to a result in [K]. ) The result in [K] is quite general, and so we will specialize it to kites. In this case, a kite is quasirational iff it is rational. , where p+q−1 Ja = Iak+i . i=0 The endpoints of the J intervals correspond to necklace orbits. A necklace orbit (in our case) is an outer billiards orbit consisting of copies of the kite touching vertex to vertex.

Our robust geometric limit argument works the same way with only notational complications. Let be the component of that tracks β. The forward direction + remains within a bounded distance of the baseline L of and yet is not periodic. Hence + travels inﬁnitely far either to the left or to the right. Since L has an irrational slope, we can ﬁnd a sequence of vertices {v n } of + such that the vertical distance from v n to L converges to ζ + N for some integer N. Let wn = v n − (0, N). Let γn be the component of n containing wn .

Proof: Recall that is the ﬁrst return map to R+ × {−1, 1}. As in our proof of the Room Lemma, ( p/q) has valence 2 at (0, 0). But ( p/q) describes the forward orbit of p = (1/q, 1) under . If some vertex of ( p/q) has valence 1, then has order 2 when evaluated at the corresponding point. But then has order 2 when evaluated at v. But then ( p/q) has valence 1 at (0, 0). This is a contradiction. ✷ 1I am grateful to Dmitry Dolgopyat and Giovanni Forni for pointing this out to me. 4 ORBIT EXCURSIONS Remark: The material in this section is not needed for the proof of the Erratic Orbits Theorem.

### An Introduction to Algebraic Geometry by K. Ueno

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