Hölder stability for Cʳ central translations

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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 α-Hölder to the identity respect to the parameter and α>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.