Birkhoff's ergodic theorem guarantees that for typical orbits in a well-behaved dynamical system, time averages converge to space averages. The orbit visits every region in proportion to its measure, and the running average settles down. This is the mathematical foundation for statistical mechanics: you can replace ensemble averages with time averages because the latter converge.
Barrientos and Chavez (arXiv:2501.00266) identify a mechanism that prevents this convergence: the arcsine law. In skew products with one-dimensional fiber dynamics, when the fiber maps' iterates behave like a random walk, the time spent above the mean follows an arcsine distribution — meaning the orbit spends most of its time either well above or well below the mean, rarely crossing. The running average doesn't settle; it keeps shifting direction.
The arcsine law is one of the most counterintuitive results in probability. In a fair coin-flip game, you'd expect the running total to cross zero frequently — to spend roughly equal time positive and negative. Instead, the arcsine distribution says the most likely outcome is spending almost all the time on one side, and the least likely outcome is the even split. The orbit looks biased even though the system is fair.
When this arcsine structure governs the fiber dynamics of a skew product, Birkhoff averages diverge for almost every orbit. Not because the system is badly behaved in the usual sense — it's just that the natural statistical structure of the underlying random walk prevents convergence. The time average never settles because the orbit keeps making long excursions to one side.
Convergence fails not from pathology but from a law.