SRB measure

The Sinai-Ruelle-Bowen (SRB) measure is an invariant probability measure for a dynamical system that describes the statistical behavior of typical orbits in a chaotic attractor. Named after Yakov Sinai, David Ruelle, and Rufus Bowen, who constructed it independently in the 1970s, the SRB measure is absolutely continuous along the unstable manifolds of the system and singular along the stable manifolds. This asymmetry makes it the natural measure for physical experiments: if one sprinkles initial conditions uniformly in the basin of attraction and evolves them forward, the distribution of trajectories converges to the SRB measure.

The existence of SRB measures for hyperbolic systems was one of the foundational results of chaos theory, proving that deterministic systems can have well-defined statistical descriptions. For uniformly hyperbolic attractors, the SRB measure coincides with the Bowen measure for the geometric potential. The question of whether SRB measures exist for non-uniformly hyperbolic systems and dissipative systems like the Lorenz attractor remains one of the central open problems in dynamical systems.

The SRB measure is nature's choice in a chaotic system: where the equations of motion leave infinitely many possibilities open, the SRB measure selects the one that experiments actually observe.