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3
votes
2answers
244 views

Unstable Foliations

Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$. Definition. $\mathcal{F}$ is said to be a unstable ...
4
votes
2answers
278 views

Eigenvalues for toral Anosov automorphisms

It is well known that on every $d$-dimensional torus there exists linear Anosov automorphisms. My question is the following: Given $k< d$ does there exists a linear Anosov automorphism of ...
3
votes
1answer
212 views

Accumulation points of the Birkhoff average of $m$

Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by ...
4
votes
1answer
325 views

Question about an early result on the mixing of geodesic flows

Let $T_t$ be the geodesic flow on a surface $S$ of constant negative curvature, and let $M(f,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where $\langle f \rangle := \int_S f(x) d\mu(x)$ and ...
11
votes
0answers
384 views

Codimension 2 foliations on simply connected 4-manifolds

Are there examples of codimension 2 foliations on simply connected compact 4-manifolds such that Every leaf is diffeomorphic to $\mathbb R^2$ Every leaf is dense? Same question for 5-manifolds ...
5
votes
1answer
280 views

Existence conditions for twisted cohomological equations?

Let $T: X \to X$ be an Anosov diffeomorphism. Suppose $f: X \to \mathbb{R}$ is Holder continuous (say with exponent $\alpha$). The question arises as to when $f$ can be written as $g \circ T - g $ ...
6
votes
2answers
380 views

Kalinin's formulation of the Anosov closing lemma

I'm trying to read a paper of Boris Kalinin on the cohomology of dynamical systems for a project. The material is geared towards topologically transitive Anosov diffeomorphisms (which is how the ...
6
votes
3answers
754 views

When is an Anosov diffeomorphism mixing?

Let $M$ be a compact Riemannian manifold and let $T : M \rightarrow M$ be Anosov. I have read here that it is an open problem to prove that $T$ is topologically mixing if $M$ is connected. Katok and ...
5
votes
1answer
520 views

Spectrum of a generic integral matrix.

My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms. These are given by integral matrices A with determinant 1 and without eigenvalues on the unit ...