Arcsine Laws and Historical Behavior of Birkhoff Averages in Skew Products
A framework in ergodic theory explaining when the time (Birkhoff) averages of an observable fail to converge along almost every orbit, called historical behavior. For skew products with one-dimensional fiber dynamics, suitable ergodic conditions make a weak form of the arcsine law force this non-convergence, tying historical behavior to the arcsine distribution of the orbits. Students learn how a probabilistic law about random walks translates into a purely dynamical statement about divergent time averages.
2501.00266
Studies 'historical behavior' in dynamical systems: an orbit is historical when the time (Birkhoff) average of a continuous observable fails to converge, as opposed to predictable orbits whose averag…