In this paper we extend the novel approach to discrete Painlevé equations initiated in our previous work [2]. A classification scheme for discrete Painlevé equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from ℙ^1×ℙ^1 by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in ℙ^3. A discrete Painlevé equation is viewed as an autonomous transformation of ℙ^3 that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in ℙ^3. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial Δ(λ) of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlevé map corresponds to a translation on the universal cover of the Riemann surface of √(Δ(λ)), rather than to a Möbius transformation of the pencil parameter λ as in [2].
We prove that the classification of real-analytic vector fields on the two-torus up to orbital topological equivalence does not admit a complete numerical invariant that is a Borel function. Moreover, smooth vector fields that are difficult to classify appear in generic smooth 7-parameter families. In dimension 2, this improves the recent result of Gorodetski and Foreman (arXiv:2206.09322) for non-classifiability of smooth diffeomorphisms up to continuous conjugacy.
