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.

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.

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. Therefore, $$ E^{u}(x)=\bigcap_{n \geq 1} D_{f^{-n}(x)} f^{n}\left(\mathscr{C}_{f^{-n}(x)}\right) ,$$ where $\{C_{x}\}_{x\in M}$ 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.

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.