Bowen measure

The Bowen measure is the unique equilibrium state for a hyperbolic dynamical system with respect to a Hölder-continuous potential, constructed by Rufus Bowen as the limit of measures supported on periodic orbits. It is the measure that maximizes the variational principle in the thermodynamic formalism, and for the zero potential it coincides with the SRB measure. Bowen proved that this measure has exponential decay of correlations and satisfies the central limit theorem, making it the natural probability distribution for studying the statistical properties of chaos.

The Bowen measure is not merely a mathematical construct; it is the measure that an experimenter would observe when measuring a chaotic system over long times. Its existence and uniqueness for hyperbolic systems was one of the foundational results that transformed chaos theory from a qualitative description of sensitivity into a quantitative statistical mechanics.

The Bowen measure is the signature of a hyperbolic system: where entropy measures how much the system forgets, the Bowen measure specifies what it remembers.