  We consider the class of diffeomorphisms of a manifold that its differential
keeps invariant a one-dimensional subbundle $E$. For that type of
diffeomorphisms is naturally defined a one-parameter family called
$E-$translation. We prove that if a diffeomorphisms in above mentioned class is
conjugate to its $E-$translation and the conjugacy is at distance
$\alpha$-H\"older to the identity respect to the parameter and $\alpha>1/2$,
then the $E$-direction is hyperbolic.
  This theorem is also sharp as it is be discussed with some examples. We also
deal with the continuously stable case in the Skew-Products context with
one-dimensional fibers, requiring extra hypothesis along the fibers like either
non-negative second derivative or negative Schwartzian.
