Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part

Crisostomo, Jorge; Tahzibi, Ali

doi:10.48550/arxiv.1604.07220

Cited by 1 publication

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“…These papers deal with a global classification of the measure, but another way to tackle the measure rigidity is the study of its disintegration along certain dynamically relevant invariant structures. This method has been proven to be very efficient to obtain strong conclusions on the dynamics [30,23,18,28,29,24,37,36,35,4,39,38,14]. Now, let us see a model which has a distinct behavior from those above.…”

Section: Introductionmentioning

confidence: 99%

Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations

Ponce¹,

Varão²

2018

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In this paper, given a Borel action G X, we introduce a new approach to obtain classification of conditional measures along a G-invariant foliation along which G has a controlled behavior.Given a Borel action G X over a Lebesgue space X we show that if G X preserves an invariant system of metrics along a Borel lamination F, which satisfy a good packing estimative hypothesis, then the ergodic measures preserved by the action are rigid in the sense that the system of conditional measures with respect to the partition F are the Hausdorff measures given by the metric system or are supported in a countable number of boundaries of balls. The argument we employ does not require any structure on G other then second-countability and no hyperbolicity on the action as well. Our main result is interesting on its own, but to exemplify its strength and usefulness we show some applications in the context of cocycles over hyperbolic maps and to certain partially hyperbolic maps. Contents 1. Introduction 1 2. Preliminaries on measure theory 6 3. Borel laminations and metric systems 9 4. Disintegration over a Borel lamination 11 5. Proof of the main Theorems 15 6. Some applications in smooth dynamics 25 References 34

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