Hyperbolic Map

A linear Map $\Bbb{R}^n$ is hyperbolic if none of its Eigenvalues has modulus 1. This means that $\Bbb{R}^n$ can be written as a direct sum of two $A$-invariant Subspaces $E^s$ and $E^u$ (where $s$ stands for stable and $u$ for unstable). This means that there exist constants $C>0$ and $0<\lambda<1$ such that

$$\vert\vert A^n v\vert\vert\leq C\lambda^n\vert\vert v\vert\vert\quad{\rm if\ } v\in E^s$$

$$\vert\vert A^{-n} v\vert\vert\leq C\lambda^n\vert\vert v\vert\vert\quad{\rm if\ } v\in E^u$$

for $n=0$, 1, ....

See also Pesin Theory