Unstable topological entropy in mean u-metrics for partially hyperbolic systems

In this paper, we give the definition of unstable topological entropy in mean u-metrics (the mean metrics in the unstable manifold) for partially hyperbolic systems. By establishing Katok's entropy formula of unstable metric entropy in mean u-metrics, we prove that the new unstable topological entropy is equal to the unstable topological entropy defined in Bowen u-metrics (the Bowen metrics in the unstable manifold). Finally, we obtain the variational principle related to the unstable topological entropy defined in mean u-metrics and unstable metric entropy, which states that the unstable topological entropy defined in mean u-metrics is the supremum of the unstable metric entropy taken over all invariant measures.