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We consider the class of diffeomorphisms of a manifold that its differentialkeeps invariant a one-dimensional subbundle $E$. For that type ofdiffeomorphisms is naturally defined a one-parameter family called$E-$translation. We prove that if a diffeomorphisms in above mentioned class isconjugate 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 alsodeal with the continuously stable case in the Skew-Products context withone-dimensional fibers, requiring extra hypothesis along the fibers like eithernon-negative second derivative or negative Schwartzian.

H\"older stability for $C^r$ central translations

H\"older stability for $C^r$ central translations