@inbook{10.2307/j.ctt7t77g.12, ISBN = {9780691142494}, URL = {http://www.jstor.org/stable/j.ctt7t77g.12}, abstract = {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}, bookauthor = {Richard Evan Schwartz}, booktitle = {Outer Billiards on Kites (AM-171)}, pages = {85--92}, publisher = {Princeton University Press}, title = {The Strip Functions}, year = {2009} }