Three problems on absense of Anosov diffeos

January 12, 2018

I would like to pose three problems on absence of Anosov diffeomorphisms on various types of manifolds. All these problems seem to be difficult and, I think, solutions must yield some new conceptual understanding of Anosov diffeomorphisms.

Problem 1 Show that {\mathbb S^3\times\mathbb S^3} does not support  Anosov diffeomorphisms.

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Partially hyperbolic diffeomorphism by surgery

December 28, 2017

In this posting I will explain a surgery construction of a weakly partially hyperbolic diffeomorphisms on a closed 3-dimensional manifold. A diffeomorphism {f\colon M\rightarrow M} is weakly partially hyperbolic if it admits an invariant dominated splitting {E^{cu}\oplus E^{s}}, where {E^s} is uniformly contracting. The following is the context.

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Codimension 2 Anosov automorphism on a nilmanifold

December 6, 2017

If {N} is a simply connected Nilpotent Lie group and {\Gamma\subset N} is a cocompact lattice then the coset space {M=N/\Gamma} is a compact nilmanifold (and every compact nilmanifold arises in this way by work of Mal’cev). An automorphism of {N} which leaves {\Gamma} invariant induces a nilmanifold automorphism. This automorphism is Anosov if and only if the corresponding automorphism of the Lie algebra {\mathfrak n} is hyperbolic. Codimension of Anosov automorphism is the minimum {k=\min(\dim E^s,\dim E^u)} and then the signature is {(k, n-k)}, where {n=\dim M}.

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Spectra of flat tori

November 27, 2017

Given a closed Riemannian manifold {M} there are several types of spectra which one can associate to it: Laplace spectrum, length spectrum and marked length spectrum. All spectra are isometry invariants and, classically, one asks how much geometry of the manifold can be recovered from the spectrum (which goes back at least to Hermann Weyl and was popularized by Mark Kac). Here I will review some well-known results on inverse spectral problem for flat tori. This blog post is based on articles of Mark Kac 1, John Milnor 2, Carolyn Gordon 3 and Robert Brooks 4, and the book of Conway and Sloane. However all mistakes are my sole responsibility.

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Global structural stability

November 11, 2017

Global structural stability (GSS) is known in two prominent setups: Anosov diffeomorphisms on tori and flows in negative curvature. For both of these setups there are two different approaches to establishing GSS:

  1. Local to global: use local structural stability to extend conjugacies alongs paths.
  2. Through dynamics at infinity: construct the conjugacy by going to induced dynamics at infinity and then back.

In low dimension these approaches yield exactly the same results. However in higher dimensions and in partially hyperbolic setting the two approaches have variable efficiency. After recalling the main theorems I will focus discussion on these aspects.

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Three problems on Anosov bundles

November 3, 2017

I would like to pose three problems on Anosov bundles, all of which seem to be difficult. I will stick to the case when the fiber is a torus, even though some questions can be posed for more general fibers.

To follow are the definitions, the problems and then some remarks.
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Generalized Verjovsky Conjecture.

September 28, 2017

Here is an open problem on higher dimensional Anosov flows, which comes from our joint work with C. Bonatti, T. Barthelme and F. Rodriguez Hertz.

Does there exist an Anosov flow which is not orbit equivalent to a suspension of an Anosov diffeomorphism and whose stable and unstable distributions have different dimensions?

 

3 p.h. problems

October 10, 2009

This account is mainly for participating in the partially hyperbolic book project.

Let me post 3 questions to which I would like to learn the answers.

1) Is there an example of p.h. diffeo on a simply connected manifold?

2) Let f be a coherent p.h.d. Is the following true?

There exists a central leaf W^c(x) such that for any other central leaf W^c(y) there exists a covering map p:W^c(x)\to W^c(y) defined as composition of holonomies.

There are interesting examples that illustrate this scenario. For example, W^c(x) can happen to be a surface of infinite genus.

3) Let L:\mathbb T^3\to\mathbb T^3 be a hyperbolic automorphism with real spectrum \lambda_1<1<\lambda_2<\lambda_3. Let f be a small perturbation of L. Then there is one dimensional f-invariant foliation W^{su} with expansion rate \lambda_3\pm\varepsilon. Is W^{su} minimal/transitive?