a Lipschitz map $\widetilde{\beta}_b:M \to B$ such that $$\label{eqn:fibers}
d(\widetilde{\beta}_b,b)_{C^0} \leq O(\rho), \textrm{ }\textrm{ }\textrm{ }\mLip \widetilde{\beta}_b \leq O(\rho)$$

$$\label{eqn:hol}
d(\widetilde{\beta}_b,\widetilde{\beta}_{b'})_{C^0} \le \frac {d(b , b' )^{\a-O(\rho)}}{O(\rho)^\alpha},$$

where $$\label{eqn:defalpha}
\alpha=\min\left(\frac {\ln \lambda}{\ln \lambda_-}, \frac {\ln
\mu}{\ln \mu_-}\right).$$ Moreover, the map $H^{-1}$ is also continuous, with the same $\alpha$.

Let us first make a remark about the exponent $\alpha$. In many cases (e.g. when $h$ is the solenoid map or a linear Anosov diffeomorphism of a torus), it may happen that $\lambda_-=\lambda$ and $\mu_-=\mu$. In that case, in the above theorem we have $\alpha=1$, and thus the exponent can be made arbitrarily close to 1 by making $\rho$ small enough.

If $\Lambda=B$ (which would require $h$ to be surjective) the invariant set $Y$ equals the entire phase space $X$. This may be proven in similar fashion to Proposition \[prop:whole phase space\] below.

Let us explain the usefulness of this Theorem. Quite often, one may use skew products $\F$ to exhibit various dynamical or ergodic phenomena (see [@GI99], [@GI00], [@GIKN], [@IKS08], [@DG]). One would like to prove the same properties for small perturbations $\G$ of $\F$, but $\G$ is a priori not a skew product anymore. However, letting $G=H\circ \G|_Y\circ H^{-1}$, statement 