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monograph guide


A kite is a convex quadrilateral with an axis of bilateral symmetry that goes through a pair of opposite vertices. (See Figure below.) Every kite is affinely equivalent to a kite K(A) with vertices (0,1); (0,-1); (-1,0); (A,0) with A in (0,1). The kite is called (ir)rational if A is (ir)rational. Outer billiards is an affinely natural dynamical system, so the study of outer billiards on kites reduces to the study of outer billiards on K(A).

the outer billiards map

The outer billiards map on K is x0-->x1. The line segment from x0 to x1 is bisected by a vertex of K, and a person walking from x0 to x1 sees K on the right. Iterating the o.b.m. generates the forwards orbit x0-->x1-->x2-->x3... The backwards orbit is defined using the inverse o.b.m. Both directions are defined on a full measure subset of R^2-K. We call the map o2(x0)=x2 the square map .

kinds of orbits

Reflection in each vertex of K(A) preserves the family of horizontal lines having height an odd integer. These are the blue lines in the applet at right. Hence, any orbit that starts in this set of lines remains there. We call such an orbit special We call an outer billiards orbit erratic if both the forwards and backwards orbits are unbounded, and also enter into every neighborhood of the vertex set of K. Erratic orbits oscillate in the widest possible sense.


The demo at the right lets you see up to 20 iterates of the o.b.m. on K(3/7). (We just picked a random example.) Use the middle mouse button to select points on the big window. Use the left/right mouse buttons to scale the picture. If you don't have a 3 button mouse, or if yours is not working properly with the applet, you can use the mouse emulator at the top left. The applet only lets you plot special orbits. Use the arrow keys to select the number of iterates. max=20. Toggle the two buttons to see the o.b.m. and/or the square of o.b.m.