Let $M$ be a smooth manifold, $U \subset M$ an open set, $f : U \to M$ a $C^1$ diffeomorphism onto its image and $\Lambda \in U$ a hyperbolic set for $f$. 


Fix a sufficiently small $\gamma > 0$ and consider a $(\lambda, \mu)$-splitting and the family of horizontal cones
$$H\_x^\gamma = \{ u + v : E^u\_x, v\in E^s\_x, \| v \| \leq \gamma \; \| u \| \}.$$


Can someone give me a hint on how to show that 
$$Df\_x H\_x^\gamma \subset H\_{f(x)}^{\lambda \mu^{-1} \gamma}?$$


P.S. This is a part of the proof for Proposition 6.4.6 from "Introduction to the Modern Theory of Dynamical Systems" - A. Katok & B. Hasselblatt.


Thank you!



Showing that $Df_x H_x^\gamma \subset H_{f(x)}^{\lambda \mu^{-1} \gamma}$, where $H_x^\gamma$ is a family of horizontal cones
Since $E^u, E^s$ are equivariant, the projections of $df\_x (u + v)$ according to $E^u\_{f x}, E^s\_{fx}$ are exactly $df\_x u$ and $df\_x v$, respectively. So,
$$
\frac{\| df\_x v \|}{ \|df\_x u\|} \leq \frac{\| df\_x|\_{E^s\_x}\|}{m( df\_x|\_{E^u\_x})} \frac{\|v\|}{\|u\|} \, ,
$$
where $m(A) = \min \{ \| A v \| : \| v\| =1\} = \| A^{-1} \|^{-1}$ is the minimal norm of a linear operator. The desired cone membership follows from this estimate.


