\right | }_{C^0(X)}, \ {\left| \left| \frac {\partial f_b^{\pm
1}}{\partial m}\right | \right | }_{C^0(X)}\right)=:L <
\min(\lambda^{-1}, \mu^{-1}).$$

Skew products are very useful in constructing dynamical systems with various properties. However, one often wants to study generic phenomena of dynamical systems, and therefore one also has to study small perturbations of skew products.

\[def:rpert\] Given $\rho>0$, a *$\rho-$perturbation* of the skew product is a $C^1-$morphism $\G:X \rightarrow X$ such that $$\label{eqn:perturbation} d(\G^{\pm 1}, \F^{\pm 1})_{C^1(X)}< \rho.$$

Let us make a notational convention. In this paper, we will consider a fixed skew product $\F$ and a neighborhood $\Omega \ni \F$ in the $C^1-$norm. We will often be concerned with small perturbations $\G \in \Omega$ of $\F$, and with various geometric objects related to these perturbations (such as central foliations, Hölder exponents etc). The leaves of central foliations of the perturbed maps $\G$ are graphs of parameter dependent maps $\beta_b$, or in other words, parameter dependent perturbations of the central fibers of $\F$. Whenever we write $||\beta_b|| = O(\rho)$, we mean that there exists a constant $C$ depending only on $\Omega$, such that for any $\rho-$perturbation $\G \in \Omega$ the maps corresponding to all central leaves satisfy the inequality $||\beta_b||\leq C\rho$. Thus 