Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting Part 3: full formula with arbitrary multiplicities

Abstract

We establish the full Ledrappier-Young entropy formula for ergodic invariant measures of $C^1$ diffeomorphisms when the positive Oseledets bundles and their nonpositive complement form a dominated splitting. We prove transverse formulas under three-block domination and equality of metric and unstable entropy under positive-nonpositive domination. To complete the proof, we develop a new mechanism for estimating partial entropy through comparisons of conditional information along orbits. Applications characterize vanishing transverse dimension, give upper pointwise dimension bounds and exact dimensionality of hyperbolic measures. In the quasi-isometric-center setting, this implies a $C^1$ invariance principle for measures with nonpositive center Lyapunov exponents.