On the definition of Anosov diffeomorphism
$begingroup$
I have a question about the definition of Anosov diffeomorphism. It might sound a little bit silly, but anyway.
In most definitions I saw, we say that $f: M to M$ is an Anosov diffeomorphism if there exists a decomposition $T_x M = E^s(x) bigoplus E^u(x)$, a constant $C > 0$ and $0< lambda < 1$ such that $forall v in E^s(x), forall n geq 0 : Vert d_xf^n v Vert leq C lambda^{n} Vert v Vert$ ... My question is : by $ d_xf^n v$, do we mean $d_x(f^n)$ (that is, the differiential of $f^n$ at $x$ , or $(d_xf)^n$, that is the differential of $f$ at $x$ composed with itself $n$-times. What difference does it make?
Thanks a lot !
dynamical-systems
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
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add a comment |
$begingroup$
I have a question about the definition of Anosov diffeomorphism. It might sound a little bit silly, but anyway.
In most definitions I saw, we say that $f: M to M$ is an Anosov diffeomorphism if there exists a decomposition $T_x M = E^s(x) bigoplus E^u(x)$, a constant $C > 0$ and $0< lambda < 1$ such that $forall v in E^s(x), forall n geq 0 : Vert d_xf^n v Vert leq C lambda^{n} Vert v Vert$ ... My question is : by $ d_xf^n v$, do we mean $d_x(f^n)$ (that is, the differiential of $f^n$ at $x$ , or $(d_xf)^n$, that is the differential of $f$ at $x$ composed with itself $n$-times. What difference does it make?
Thanks a lot !
dynamical-systems
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
$endgroup$
add a comment |
$begingroup$
I have a question about the definition of Anosov diffeomorphism. It might sound a little bit silly, but anyway.
In most definitions I saw, we say that $f: M to M$ is an Anosov diffeomorphism if there exists a decomposition $T_x M = E^s(x) bigoplus E^u(x)$, a constant $C > 0$ and $0< lambda < 1$ such that $forall v in E^s(x), forall n geq 0 : Vert d_xf^n v Vert leq C lambda^{n} Vert v Vert$ ... My question is : by $ d_xf^n v$, do we mean $d_x(f^n)$ (that is, the differiential of $f^n$ at $x$ , or $(d_xf)^n$, that is the differential of $f$ at $x$ composed with itself $n$-times. What difference does it make?
Thanks a lot !
dynamical-systems
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
$endgroup$
I have a question about the definition of Anosov diffeomorphism. It might sound a little bit silly, but anyway.
In most definitions I saw, we say that $f: M to M$ is an Anosov diffeomorphism if there exists a decomposition $T_x M = E^s(x) bigoplus E^u(x)$, a constant $C > 0$ and $0< lambda < 1$ such that $forall v in E^s(x), forall n geq 0 : Vert d_xf^n v Vert leq C lambda^{n} Vert v Vert$ ... My question is : by $ d_xf^n v$, do we mean $d_x(f^n)$ (that is, the differiential of $f^n$ at $x$ , or $(d_xf)^n$, that is the differential of $f$ at $x$ composed with itself $n$-times. What difference does it make?
Thanks a lot !
dynamical-systems
dynamical-systems
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
asked Jan 2 at 14:08
Noam EluarNoam Eluar
154
154
add a comment |
add a comment |
1 Answer
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$df_x:T_xMrightarrow T_{f(x)}M$, so you cannot compose $(df_x)^n$ unless $f(x)=x$.
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
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$begingroup$
$df_x:T_xMrightarrow T_{f(x)}M$, so you cannot compose $(df_x)^n$ unless $f(x)=x$.
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
$endgroup$
add a comment |
$begingroup$
$df_x:T_xMrightarrow T_{f(x)}M$, so you cannot compose $(df_x)^n$ unless $f(x)=x$.
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
$endgroup$
add a comment |
$begingroup$
$df_x:T_xMrightarrow T_{f(x)}M$, so you cannot compose $(df_x)^n$ unless $f(x)=x$.
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
$endgroup$
$df_x:T_xMrightarrow T_{f(x)}M$, so you cannot compose $(df_x)^n$ unless $f(x)=x$.
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
answered Jan 2 at 14:22
Tsemo AristideTsemo Aristide
57.2k11444
57.2k11444
add a comment |
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