By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

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N. Since T is an irrational rotation, all these points are distinct and there are n + 1 of them. To describe the initial n-segments of the cutting sequences, start with the line through the origin (0, 0) and parallel translate it along the diagonal of the unit square toward point (−1, 1). The n-segments of the cutting sequence change when the line passes through a vertex of one of the first n ladders. As we have seen, there are n + 1 such events, and hence p(n) = n + 1. 13. One can similarly encode billiard trajectories in a k-dimensional cube: the cutting sequence consists of k symbols corresponding to the directions of the faces.

The curve γ is a part of an ellipse with foci F1 and F2 ; the curve Γ is a parabola with focus F2 . 3 53 4. 2. Trap for a beam of light that a vertical ray, entering the curve through a window, will tend to the major axis of the ellipse and will therefore never escape. The next question foreshadows Chapter 7: can one construct a compact trap for the set of rays sufficiently close to a given ray, that is, making small angles with it? 1 for the answer. The construction of an ellipse with given foci has a parameter, the length of the string.

Xn−1 that lie in I. 1) lim n→∞ |I| k(n) = n 2π for every I. 1. 3. If θ is π-irrational, then the sequence xn = x + nθ mod 2π is equidistributed on the circle. Proof. (Sketch). 2) 1 n→∞ n n−1 lim f (xj ) = j=0 1 2π 2π f (x)dx; 0 23 2. Billiard in the Circle and the Square the time average equals the space average. To deduce equidistribution one takes f to be the characteristic function of the arc I, equal to 1 inside and 0 outside. 1). One may approximate the function f (x) by a trigonometric polynomial, a linear combination of cos kx and sin kx with k = 0, 1, .