Abstract
We study Birkhoff averages along trajectories of smooth reparameterizations of irrational linear flows of the two torus with two stopping points, say p and q, of quadratic order. The limiting behaviour of such averages is independent of the starting point in a set of full Haar-Lebesgue measure and depends in an intricate way on the Diophantine properties of both the slope α of the linear flow as well as the relative position of p and q. In particular, if α is Diophantine, then Birkhoff limits diverge almost everywhere (historic behaviour) and if α is sufficiently Liouville, then there exists some p and q such that the Birkhoff averages converge almost everywhere (unique physical measure).
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