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Historic behaviour vs. physical measures for irrational flows with multiple stopping points. (English) Zbl 1510.37009

Summary: 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 \(\alpha\) of the linear flow as well as the relative position of p and q. In particular, if \(\alpha\) is Diophantine, then Birkhoff limits diverge almost everywhere (historic behaviour) and if \(\alpha\) is sufficiently Liouville, then there exists some p and q such that the Birkhoff averages converge almost everywhere (unique physical measure).

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A44 Relations between ergodic theory and number theory
37E35 Flows on surfaces
11K50 Metric theory of continued fractions
11J71 Distribution modulo one
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