Can quantum dynamics emerge from classical chaos?

Anosov geodesic flows are among the simplest mathematical models of deterministic chaos. In this survey we explain how, quite unexpectedly, quantum dynamics emerges from purely classical correlation functions. The underlying mechanism is the discrete Pollicott Ruelle spectrum of the geodesic flow, revealed through microlocal analysis. This spectrum naturally arranges into vertical bands; when the rightmost band is separated from the rest by a gap, it governs an effective dynamics that mirrors quantum evolution.