A Question of Norton–Sullivan in the Analytic Case. (23rd June 2020) - Wang, Jian; Yang, Hui
Abstract: In 1996, A. Norton and D. Sullivan asked the following question: If f:T^2→ T^2 is a diffeomorphism, h:T^2→ T^2 is a continuous map homotopic to the identity, and h f=T_ρ h, where ρ R^2 is a totally irrational vector and T_ρ :T^2→ T^2, zz+ρ is a translation, are there natural geometric conditions (e.g., smoothness) on f that force h to be a homeomorphism? In [ 22 ], the 1st author and Z. Zhang gave a negative answer to the above question in the C^∞ category: in general, not even the infinite smoothness condition can force h to be a homeomorphism. In this article, we give a negative answer in the C^ω category (see also [ 22, Question 3]): we construct a real analytic conservative and minimal totally irrational pseudo-rotation of T^2 that is semi-conjugate to a translation but not conjugate to a translation.