I'm working on Uniformly hyperbolic finite-valued $SL(2,R)$ -cocycles( article from Arthur Avila and Jairo Bochi)and at the beginning of my researches,i want to know the exactly meaning of the title of this Article, so i started to study the book " Nonuniform-hyperbolic" from Pesin and Barreira .In page 12 , there is the first definition of uniformly hyperbolic and non-uniformly hyperbolicity( i think its about uniformly hyperbolic points) . It says one of the conditions for making this definition is :" the angle between local stable and local unstable manifolds which was mentioned at point $p$ is Uniformly bounded away from zero in $p$." .Actually this is the sentence which i don't have any idea what it means and want to help me to understand it since i really need to have a right view of the paper title and after that start my researches in it.
If there exist any free available paper or book which gives easier definition of mentioned tools please let me know and i appriciate your help anyway.
Thank you in advance
It is essentially just linear algebra once you equip your manifold with a Riemannian metric. The question boils down to defining the angle $\angle(U, V)$ between two linear subspaces of a euclidean vector space $W$ (a Hilbert space if you like). Then $$ \angle(U, V)= \inf_{u, v} \angle(u, v) $$ where the infimum is taken over all nonzero vectors $u\in U, v\in V$. Now, if $T^s_p, T^u_p\subset T_pM$ are tangent spaces to stable/unstable manifolds then you get the number $$ \alpha(p)= \angle(T^s_p, T^u_p)$$ Uniform hyperbolicity requires $$ \inf_{p\in M} \alpha(p) >0. $$ The non-uniform hyperbolicity is more complicated.
Incidentally, it is Pesin nor Persin.