Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting Part 1: Unstable entropy formula and invariance principle

Abstract

We study the unstable entropy of $C^1$ diffeomorphisms with dominated splittings. Our main result shows that when the zero Lyapunov exponent has multiplicity one, the center direction contributes no entropy, and the unstable entropy coincides with the metric entropy. This extends the celebrated work of Ledrappier-Young [18] for $C^2$ diffeomorphisms to the $C^1$ setting under these assumptions. In particular, our results apply to $C^1$ diffeomorphisms away from homoclinic tangencies due to [20]. As consequences, we obtain several applications at $C^1$ regularity. The Avila-Viana invariance principle [7, 33] holds when the center is one-dimensional. Results on measures of maximal entropy due to Hertz-Hertz-Tahzibi-Ures [25], Tahzibi-Yang [33], and Ures-Viana-Yang-Yang [34, 35] also remain valid for $C^1$ diffeomorphisms.