Hölder stability for $C^r$ central translations

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.

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