1
$\begingroup$

Let $f:M \to M$ be a $C^{2}$-Anosov diffeomorphism. Therefore, there exists an invariant splitting of the tangent bundle $T_{x}M = E^s(x) \oplus E^u(x)$ into a stable and an unstable directions, that is with $\|Df|E^s \|_{\infty} < 1$ and $\|Df^{−1}|E^u\|_{\infty}<1$. Domination can be characterized in terms of existenceof invariant cone fields. $e^{u}, e^{s}:M \to \mathbb{P}$ is the invariant directions forming the dominated splitting, i.e., $$ \{e^{u}(x)\}=\bigcap_{n \geq 1} D_{f^{-n}(x)} f^{n}\left(\mathscr{C}_{f^{-n}(x)}\right) ,$$ where $\{\mathscr{C_{x}}\}_{x\in M}$ is a family of cone field.

As far s I know,$x\mapsto e^{u}(x)$ is continuous. I want to know that under what assumptions $x\mapsto e^{u}(x)$ is an injective map? Thanks in advance.

Improve this question
$\endgroup$
4
  • $\begingroup$ What do you mean by injective? $x \mapsto E^u(x)$ is a section of a bundle, so for $y\neq x$ belongs to different spaces. However, for certain linear cocycles the question makes sense (e.g. Anosov representations have a property of having some sort of injectivity of the bundles). $\endgroup$
    – rpotrie
    Commented Feb 28, 2021 at 14:42
  • $\begingroup$ @rpotrie : Sorry, I should have written better my question. Exactly, I want to ask the question for linear cocycles. Could you please explain to me which linear cocycles have such properties? or at least introduce some references. Thanks in advance. $\endgroup$
    – Adam
    Commented Feb 28, 2021 at 15:10
  • $\begingroup$ While it is a somewhat artificial condition for general linear cocycles (it is not stable under conjugacy, for instance), it does make sense in some geometric contexts. You may find arxiv.org/abs/2002.07015 and references therein useful. $\endgroup$
    – rpotrie
    Commented Feb 28, 2021 at 15:20
  • $\begingroup$ @rpotrie Thanks! $\endgroup$
    – Adam
    Commented Feb 28, 2021 at 15:24

0

Reset to default

You must log in to answer this question.