Let $M$ be a Riemannian manifold, $U \subset M$ be open, $f: U \to M$ a $C^1$ diffeomorphism. Let $\Lambda \subset U$ be compact and $f$-invariant, $\lambda \in (0,1)$ and $C > 0$, and a family of subspaces $E^s(x) \subset T_x M$ and $E^u (x) \subset T_x M$. Then one of the requirements of $\Lambda$ to be hyperbolic, is the following: $$ \| df_x^n v^s\| \leq C \lambda^n \|v^s\| $$ for every $v^s \in E^s(x)$. I am wondering what the notation $d f_x^n v^s$ means in this context. Is it the derivative (Jacobian) of $f^n$ evaluated in $x$ and applied to $v^s$?
What does $d f_x^n v^s$ mean?
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dynamical-systems
1 Answers
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The map $$ d_xf\colon T_xM\to T_{f(x)}M $$ (or with your notation $df_x$) is defined by $$ d_xf \gamma'(0)=(f\circ\gamma)'(0) $$ for any differentiable curve $\gamma$ with $\gamma(0)=x$. The map $d_xf^n$ is defined in a similar manner.
When $M=\mathbb R^n$ the map $d_xf^n$ is indeed the Jacobian matrix (you should write "Jacobian" for the determinant of the Jacobian matrix).