Variational principles for sub-additive unstable topological pressures on saturated subsets

Abstract

In this paper, under the assumption of unstable almost product property for C 1-smooth partially hyperbolic diffeomorphisms, we establish variational principles for their unstable Bowen topological pressures and unstable packing topological pressures on the typical saturated subsets G K , where K is a given non-empty compact connected set of invariant measures. Actually, we show that these two dimension-like topological quantities coincide with the infimum and supremum respectively of the summation of unstable metric entropy and Lyapunov exponent, where the infimum and supremum are taken over all measures inside K. Besides, we also show that G K has full unstable topological capacity pressure for reasonable sub-additive potentials.