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

Question

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!

Answer 1

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.