Paths in phase space
The previous problem makes it clear that our billiard table is deterministic, in that if you know the position and velocity of a ball at one time you can predict the motion of the ball for all time. In reality of course it is impossible to exactly specify the position and velocity of a ball, there is always some measurement error. For many systems, the measurement error doesn't overly matter as the error does not grow with time. Hence if you are within some tolerance to start, you will remain within that tolerance.
For example, consider our ball again. Now, however, instead of knowing that the ball starts exactly at the origin, we only know that it starts somewhere nearby the origin, i.e. within a circle of radius \(\epsilon\) of the origin. However, it still has velocity (0,1) to start. The ball bounces off the wall N times. After 4N seconds, the ball is within a circle of radius L of the origin. What is L?