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

TL;DR

This paper establishes that for certain diffeomorphisms with invariant subbundles, a Hölder conjugacy to an $E$-translation implies hyperbolicity of the $E$-direction, with sharpness demonstrated through examples.

Contribution

It proves a sharp Hölder stability criterion for $C^r$ central translations, linking conjugacy regularity to hyperbolicity in diffeomorphisms with invariant subbundles.

Findings

Hölder conjugacy with exponent > 1/2 implies hyperbolicity of the $E$-direction.

The stability result is sharp, supported by explicit examples.

Extension of the stability analysis to skew-product systems with additional fiber hypotheses.

Abstract

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.