Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting Part 2: Entropy formulas and measure dimension

Abstract

In this paper, we partially extend the Ledrappier-Young entropy formula to invariant measures of $C^1$ diffeomorphisms with dominated splittings. For such measures, we show that whenever the $i$-th Lyapunov exponent has multiplicity one, the $i$-th transverse entropy equals the product of the $i$-th Lyapunov exponent and the corresponding transverse measure dimension. Furthermore, if all intermediate non-negative Lyapunov exponents have multiplicity one, then the Ledrappier-Young entropy formula holds. As applications, we derive $C^1$ versions of numerous results in measure dimension theory, including the famous works by Ledrappier-Young, Barreira-Pesin-Schmeling, and Ledrappier-Xie.