Ledrappier-Young formula

The Ledrappier-Young formula is a foundational theorem in smooth ergodic theory that connects three fundamental invariants of a dynamical system: the Lyapunov exponents, the Kolmogorov-Sinai entropy, and the fractal dimension of an invariant measure. Proved independently by Francois Ledrappier and Lai-Sang Young in the mid-1980s, the formula generalizes the Pesin entropy formula by providing a dimensional interpretation of entropy in terms of the system's Lyapunov spectrum. Where Pesin's formula states that entropy equals the sum of positive exponents under strong regularity conditions, the Ledrappier-Young formula decomposes entropy dimensionally — revealing that each positive exponent contributes to entropy in proportion to the dimension of the corresponding unstable manifold.

The formula resolves a tension that plagued early chaos theory: the relationship between geometric complexity (dimension), dynamical instability (Lyapunov exponents), and information production (entropy) was known to be intimate, but the exact nature of the triad remained elusive. The Ledrappier-Young formula provides the rigorous bridge, showing that these three invariants are not merely correlated but structurally coupled through the dimension theory of invariant measures.

The Formula and Its Components

For a diffeomorphism preserving a measure μ with non-zero Lyapunov exponents almost everywhere, the Ledrappier-Young formula states:

h_μ = Σ_{i: λ_i > 0} λ_i · δ_i

where h_μ is the Kolmogorov-Sinai entropy, λ_i are the positive Lyapunov exponents, and δ_i are the partial dimensions of the measure along the corresponding unstable manifolds. Each δ_i is a number between 0 and the dimension of the manifold, measuring how spread