A dynamical system with oscillating time average

Given a discrete dynamical system T, one can ask what the time average of the system will be, that is, what is the average position of T^n(x) for large n? Birkhoff’s ergodic theorem, one of the most important results in ergodic theory, says that for an ergodic system on a finite measure space, the time average will in the limit as n -> infinity be equal to the space average for almost all initial values x. In this thesis we study time averages of a dynamical system T : [0, 1] -> [0, 1] that depends on a parameter alpha. We show that there are values of alpha for which the points x, T(x), T^2(x), ... are equally often in the right half of [0, 1] as on the left half. We also show that for other values of alpha, the time average never converges, but instead oscillates between being concentrated on the left and right halves of the unit interval. In the process, we also prove the existence of an absolutely continuous invariant ergodic measure for T. (Less)