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======  Collapsed Anosov flows. ======

Short abstract: We will present a new class of partially hyperbolic systems in dimension 3. The idea is to present them in such a way to emphasize the main properties and potential applications.

Extended abstract: Partially hyperbolic systems arise naturally when studying robust dynamical behavior such as robust transitivity and stable ergodicity. It is also a natural setting to understand statistical properties of dynamical systems beyond uniform hyperbolicity. In the recent years, many fine dynamical and statistical properties have been obtained for certain classes of partially hyperbolic diffeomorphisms (such as derived from Anosov, skew-products, or perturbations of time one maps of Anosov flows) and much effort was concentrated in showing that these settings were abundant, at least in low dimensions.

New examples have questioned this effort and until very recently were quite ill understood from a topological point of view. In recent work motivated by the classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds we have proposed a new class of partially hyperbolic diffeomorphisms that we have named 'collapsed Anosov flows'. These contain perturbations of time one maps of Anosov flows as well as all new examples and form an open and closed class of partially hyperbolic systems.

The goal of the minicourse is to introduce these systems, their properties and the tools that may help to start a systematic study of their dynamical and statistical properties (which is totally open to exploration). We note that we will not enter in the finer details of the proofs of the classification results, which are of a different nature and require much more background on the theory of 3-manifolds and foliations. We will try to remain the prerequisites to some very basic knowledge of partially hyperbolic systems (that will be recalled) and some basic hyperbolic geometry in dimension 2 (surfaces of negative curvature). The hope is that after the minicourse, people will be prepared to think about questions that have been addressed in other settings for this new class of systems.

The tentative plan of the minicourse is as follows:

- Ordered List Item Introduction and basic properties of partially hyperbolic diffeomorphisms in dimension 3. 
  -  Examples, including the most recent ones. 
  - Introduction of the class of collapsed Anosov flows. 
  - Statement of some of the main results. 
  -  Proof of some basic properties of collapsed Anosov flows. 
  - (Time permitting) Some ideas of the proof that conservative collapsed Anosov flows are ergodic.

Bibliography: 
-Basics of PH: many references, I can refer to my notes with Crovisier -http://www.cmat.edu.uy/%7Erpotrie/documentos/pdfs/Crovisier-Potrie-PH.pdf
-General topology of partially hyperbolic systems in dimension 3- Survey papers by myself and Hammerlindl as well as the one by Carrasco-Hertz-Hertz-Ures. 
- New examples: https://arxiv.org/abs/1706.04962
- Collapsed Anosov flows: https://arxiv.org/abs/2008.06547 
- Ergodicity of collapsed Anosov flows: https://arxiv.org/abs/2103.14630
- (Extra bibliography, will not be covered in the minicourse) Papers on classification: https://arxiv.org/abs/2102.02156, https://arxiv.org/abs/2008.04871 and https://arxiv.org/abs/1908.06227v3

Ideal prerequisites: 
- Basics of partially hyperbolic systems (only statements): Definitions, cone-field criteria, stable manifold theorem. Hirsch-Pugh-Shub stability.

- Standard examples: Linear maps of tori and their perturbations (DA systems, and skew-products).

- Anosov flows: Geodesic flows and suspensions, time one maps, and perturbations. If possible, some non-algebraic examples (in the course I will talk about non-algebraic examples, but emphasize on geodesic flows).

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[[ebsd2021:exemplos|Alguns exemplos
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[[ebsd2021:potrie4|Um vislumbre muito modesto de $3-$variedades
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[[ebsd2021:potrie3|Folheações Tensas e Uniformização
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[[ebsd2021:margulis|Fluxos de Anosov e Crescimento exponencial
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[[ebsd2021:potrie5|Tema 5
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[[ebsd2021:potrie6|Tema 6
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[[ebsd2021:potrie7|Tema 7
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