We study entropies caused by the unstable part of partially hyperbolicsystems. We define unstable metric entropy and unstable topological entropy,and establish a variational principle for partially hyperbolic diffeomorphsims,which states that the unstable topological entropy is the supremum of theunstable metric entropy taken over all invariant measures. The unstable metricentropy for an invariant measure is defined as a conditional entropy alongunstable manifolds, and it turns out to be the same as that given byLedrappier-Young, though we do not use increasing partitions. The unstabletopological entropy is defined equivalently via separated sets, spanning setsand open covers along a piece of unstable leaf, and it coincides with theunstable volume growth along unstable foliation. We also obtain some propertiesfor the unstable metric entropy such as affineness, upper semi-continuity and aversion of Shannon-McMillan-Breiman theorem.
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