Ledrappier-Young theory

The Ledrappier-Young formula is a foundational result in the dimension theory of dynamical systems, proved by François Ledrappier and Lai-Sang Young in the 1980s. It gives the exact Hausdorff and pointwise dimensions of an invariant measure on a non-uniformly hyperbolic attractor as a function of its Lyapunov exponents and its metric entropy. In the uniformly hyperbolic case, the formula reduces to the dimension identities that Rufus Bowen had established; in the non-uniform case, it is the only general tool that connects fractal geometry to dynamical invariants.

The formula resolves a tension that had plagued chaos theory since its inception: chaotic attractors are clearly fractal, but their dimension seemed to depend on details of the dynamics that no invariant could capture. Ledrappier and Young showed that the dimension is not arbitrary; it is determined by the competition between expansion (Lyapunov exponents) and information production (entropy), exactly the same competition that governs the thermodynamic formalism.

The Ledrappier-Young formula is the claim that a chaotic attractor's fractal structure is not a decorative curiosity but a computable property of its dynamics. To speak of dimension without speaking of entropy and Lyapunov exponents is to describe a storm without mentioning pressure or temperature.

The formula's connection to the entropy spectrum of the measure — and its behavior under stochastic perturbations — remains an active area of research.