The rigidity of pseudo-rotations on the two-torus and a question of Norton-Sullivan

Abstract

We show that under certain boundedness condition, a C r conservative irrational pseudo-rotations on T 2 with a generic rotation vector is C r−1 -rigid. We also obtain C 0 -rigidity for Hölder pseudo-rotations with similar properties. These provide a partial generalisation of the main results in [Bra15, AFLXZ15].We then use these results to study conservative irrational pseudo-rotations on T 2 with a generic rotation vector that is semi-conjugate to a translation via a semi-conjugacy homotopic to the identity. We show that the conservative centralizers of any such diffeomorphism is isomorphic to a uncountable subgroup of R 2 /Z 2 . In connection with a question of Alec Norton and Dennis Sullivan, we describe the topologically linearizable maps within this class using the topology of the conservative centralizer group. In the minimal case, we obtain a precise characterization of topological linearizability for all totally irrational vectors.We also construct a C ∞ conservative and minimal totally irrational pseudo-rotation diffeomorphism that is semi-conjugate to a translation, but is topologically nonlinearizable. This gives a negative answer to the question of Norton and Sullivan in the C ∞ category.

Periodic point free homeomorphisms and irrational rotation factors

“…For some time these were the only known positive results. At this point it is interesting to remark that Wang and Zhang have recently proved in [WZ18] the existence of a C ∞ area-preserving diffeomorphism of T 2 which is a topological extension of a rigid rotation but is not conjugate to it.…”

A question of Norton-Sullivan in the analytic case

“…In [PaSa13], A. Passeggi and M. Sambarino also mentioned the question: whether there exists r so that if f : T 2 → T 2 is a C r diffeomorphism semi-conjugate to an ergodic translation, then f is conjugate to it. For more recent developments, we mention [Kar18,Nav18,Mer18,WZ18].…”

“…In [WZ18], the first author and Zhang constructed a smooth diffeomorphism which is isotopic to the identity and semi-conjugate to a minimal translation T α , but not conjugate to T α , which is a C ∞ counter-example to the Norton-Sullivan's question. The construction in [WZ18] combined the classical Anosov-Katok method (see [AK70,FK04]) with Jäger's theorem [Jäg09] (see Theorem 2 below). In this article, we will construct a C ω counter-example to the question of Norton-Sullivan.…”

“…In this article, we will construct a C ω counter-example to the question of Norton-Sullivan. Our strategy in this paper mainly follows from the approximation by conjugation construction scheme in the proof of Theorem 4 in [WZ18]. However, as we require the map is real-analytic, we will apply certain technique of analytic approximations in [Ban17,BK18] to customize the desirable analytic conjugacies.…”

“…We give some remarks about our theorem. As the constructions in [WZ18] and in this article, we use the classical Anosov-Katok method, the rotation vector of the constructed map is Liouvillean, which is the price to pay in order to get the smoothness of the pseudo-rotation. It seems difficult to construct a pseudo-rotation as in Theorem 1 with Diophantine rotation vector (see Section 2 for the definitions).…”