Approximately transitive diffeomorphisms of the circle

Rational diffeomorphisms of the circle

We prove an interpolation theorem for rational circle diffeomorphisms: A set of N complex numbers of unit modulus may be mapped to any corresponding set by a ratio of polynomials p(z)/q(z) which restricts an invertible mapping of the unit circle S ⊂ C onto itself.

Groups of Circle Diffeomorphisms

Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms

In this article, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of α, for any diffeomorphism f of rotation number α, it is possible to accumulate Rα with a sequence hn f h−1 n , hn being a diffeomorphism. The second result is: for a Baire-dense set of α, given two commuting diffeomorphisms f and g, such that f has α for ro...

Estimates of the Linearization of Circle Diffeomorphisms

A celebrated theorem by Herman and Yoccoz asserts that if the rotation number α of a C∞-diffeomorphism of the circle f satisfies a Diophantine condition, then f is C∞-conjugated to a rotation. In this paper, we establish explicit relationships between the Ck norms of this conjugacy and the Diophantine condition on α. To obtain these estimates, we follow a suitably modified version of Yoccoz’s p...

Topological Conjugacy of Circle Diffeomorphisms

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than [5]'s Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria.