for skew products {#sub:perhol}
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Throughout this paper a *$C^r-$morphism* will refer to a $C^r$ map with a $C^r$ inverse. We will use this notion both for maps of a manifold (with or without boundary) onto itself, and for maps of a manifold with boundary strictly into itself.\
Given a linear operator $A$ on a normed linear space $V$, when we write $a\leq |A|\leq b$ we mean that $$a|v|\leq |A(v)|\leq b|v|,
\textrm{ }\textrm{ }\textrm{ }\forall v\in V.$$ We will use this convention repeatedly throughout the paper.\
Let $B$ be a compact Riemannian manifold, henceforth called the *base*. Suppose $h:B\rightarrow B$ is a $C^2-$morphism with a hyperbolic invariant subset $\Lambda \subset B$. When $h$ is onto, we can take $B = \Lambda $. When $h$ is into, we can take $\Lambda $ to be the maximal attractor of $h$: $$\Lambda = \bigcap_{n \geq 0}
h^n(B).$$ Being hyperbolic, the map $h$ will have contracting and expanding directions in the vicinity of $\Lambda$. Thus there exist a Riemannian metric $d$ on $B$ and real numbers $0\leq \lambda_-\leq
\lambda<1$ and $0\leq \mu_-\leq \mu<1$, as well as a decomposition of the tangent bundle: $$\label{eqn:splittangent}
TB|_\Lambda=E^s\oplus E^u,$$ such that $$dh:E^s
\rightarrow E^s\textrm{ }\textrm{ }\textrm{ and }\textrm{ 