Flexible periodic points

We define the notion of $\varepsilon$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\varepsilon$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that $\varepsilon$-perturbation to an $\varepsilon$-flexible point allows to change it in a stable index one periodic point whose (one dimensional) stable manifold is an arbitrarily chosen $C^1$ -curve. We also show that the existence of flexible point is a general phenomenon among systems with a robustly non-hyperbolic two dimensional center-stable bundle.