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The conjugating map for commutative groups of circle diffeomorphisms

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Abstract

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC 2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C 1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Bryna Kra

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  1. Bryna Kra
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Correspondence to Bryna Kra.

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Kra, B. The conjugating map for commutative groups of circle diffeomorphisms. Israel J. Math. 93, 303–316 (1996). https://doi.org/10.1007/BF02761108

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  • DOI: https://doi.org/10.1007/BF02761108

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