Unstable entropies and Dimension Theory of Partially Hyperbolic Systems

Abstract

In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in [8,30]. Our results give new insights to the multifractal analysis for partially hyperbolic systems.

Unstable Metric Pressure of Partially Hyperbolic Diffeomorphisms with sub-additive Potentials

“…In fact, the unstable metric entropy in [8] has root in the entropy introduced by Ledrappier and Young ( [11]), and is easier to apply. In [15], Tian and Wu generalize the above result with additional consideration of an arbitrary subset (not necessarily compact or invariant). In [9], Hu, Wu, and Zhu investigated the unstable topological pressure for additive potentials, and obtained a variational principle.…”

Sub-additive unstable topological pressure on saturated subsets

“…They were introduced and studied in [5] by Hu, Hua, and Wu; moreover, they obtained the corresponding Shannon-McMillan-Breiman theorem, as well as the corresponding variational principle. In [12], Tian and Wu generalize the result above with additional consideration of an arbitrary subset (not necessarily compact or invariant). In [6], Hu, Wu, and Zhu investigated the unstable topological pressure for additive potentials.…”

“…Remark 2.9. The definition above appeared in Section 5 of [12], and they showed that C 1 -smooth partially hyperbolic systems enjoy this property, see Theorem D there. In fact, they proved that there is a set C with µ(C) > 0 such that for any…”

“…X. Tian showed the unstable uniform separation property in [12]; actually the unstable almost product property implies that the statement could pass to general invariant measures as follows.…”

Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

“…Some months after the submission of this paper X. Tian and W. Wu made a preprint[16] where they use the expression at the right hand side of Theorem 3.4 to prove several theorems concerning the unstable entropy of non-compact subsets. Although some of the results they obtained are already contained in this paper, their results where stablished following a different approach.…”