05-431 A. GONZALEZ-ENRIQUEZ

A non-perturbative theorem on conjugation of torus diffeomorphisms to rigid rotations (463K, pdf) Dec 21, 05

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Abstract. The problem of conjugation of torus diffeomorphisms to rigid rotations is considered here. Rather than assuming that the diffeomorphisms are close to rotations, we assume that the conjugacy equation has an approximate solution. First, it is proved that if the rotation vector is Diophantine and the invariance error function of the approximate solution has sufficiently small norm, then there exists a true solution nearby. The previous result is used to prove that if an element of a family of diffeomorphisms $\{\, f_{\mu}\}_{\mu}$ is conjugated to a rigid rotation with Diophantine rotation vector, then there exists a Cantor set $\calC$ of parameters such that for each $\mu\in\calC$ the diffeomorphism $f_{\mu}$ is conjugated to a Diophantine rigid rotation with rotation vector that depends on $\mu\in\calC$ in a Whitney-smooth way.

Files: 05-431.src( 05-431.keywords , torus-maps.pdf.mm )