Unstable entropy along invariant laminations

In this paper, we study the unstable entropy along an invariant lamination for general C1+α (α > 0) diffeomorphisms. We define the unstable entropy in a classic way using finite or countable partitions and certain natural measurable partitions subordinate to the unstable manifolds inside the lamination, and show that it coincides with the Ledrappier–Young entropy defined using increasing measurable partitions. We then show that the unstable entropy map is upper semicontinuous on a set of invariant measures with the same expansive rate or with the same hyperbolic rate, the latter extending a classic result by Newhouse. For the topological aspect, we introduce a version of unstable topological entropy, which captures the complexity by those points uniformly returning to Pesin sets. In the end, the variational principle is established for unstable metric and topological entropies.