Unstable entropies and variational principle for partially hyperbolic diffeomorphisms

Abstract

We study entropies caused by the unstable part of partially hyperbolic systems. 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 the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem.

Unstable entropy and unstable pressure for partially hyperbolic endomorphisms

“…Then following the proof of Lemma 3.1 in [1], we can prove the above claim since f is uniformly expanding along W u -foliation.…”

“…Clearly η is a measurable partition of M f (cf. p34 in [1] for more details). In addition, by the definition of W u ǫ (x, f ) and Theorem 2.1, if µ(∂(Π(α))) = 0, η is a measurable partition subordinate to W u -foliation, where ∂(Π(α)) = A∈α ∂(Π(A)) and for B ⊂ M , ∂B means the boundary of B.…”

“…Then following the line of the proof for Lemma 2.11 in [1], we can prove the remaining results in Fact 1, where Fatou's Lemma and Lemma 3.4 are needed.…”

“…Via the inverse limit space, the unstable manifolds can be well defined, which implies us that the unstable entropy and unstable pressure can be introduced for endomorphisms. The concept of unstable entropy is originally introduced by Hu, Hua and Wu in [1] for partially hyperbolic diffeomorphisms, which gives a description of the complexity of a system along unstable manifolds. In [1], a complete discussion is given, including the relationship between unstable metric entropy and Ledrappier and Young's entropy, a version of Shannon-McMillan-Breiman Theorem and a variational principle relating unstable metric entropy and unstable topological entropy.…”

“…The concept of unstable entropy is originally introduced by Hu, Hua and Wu in [1] for partially hyperbolic diffeomorphisms, which gives a description of the complexity of a system along unstable manifolds. In [1], a complete discussion is given, including the relationship between unstable metric entropy and Ledrappier and Young's entropy, a version of Shannon-McMillan-Breiman Theorem and a variational principle relating unstable metric entropy and unstable topological entropy. It is important to point out that in [1] the unstable metric entropy is given by the conditional entropy of a finite partition with respect to a measurable partition, instead of the form in Ledrappier and Young's papers [4,5].…”