Examples of minimal diffeomorphisms on \(\mathbb {T}^{2}\) semiconjugate to an ergodic translation (Q2839264)

The authors construct a family of Denjoy maps of the 2-torus \(\mathbb{T}^2\) of class \(C^r\) with \(r \in [1,3)\). The main result in the paper reads as follows:NEWLINENEWLINE{Main Theorem 1.1:} For every \(r \in [1,3)\) there exists a diffeomorphism \(f : \mathbb{T}^2 \rightarrow \mathbb{T}^2 \) which is minimal, isotopic and semiconjugate (but not conjugate) to an ergodic translation. If we denote by \(h\) the semiconjugacy, then: {\parindent=6mm \begin{itemize} \item[-] Each fiber \(h^{-1}(x)\) is either a point or an arc. \item [-] \(f\) preserves a minimal and invariant foliation with one-dimensional \(C^1\) leaves. Each fiber \(h^{-1}(x)\) is contained in a leaf of this foliation. \item [-] The set \(\{ x \in \mathbb{T}^2: h^{-1}(x) \,\,\text{is a point}\}\) has full Lebesgue measure. NEWLINENEWLINE\end{itemize}} As a consequence, \(f\) has zero entropy, is sensitive to initial conditions, point distal, non-distal and uniquely ergodic. Furthermore, \(f\) can be constructed so that there are uncountably many points \(x\) such that \(h^{-1}(x)\) is a nontrivial arc exhibiting Li-York chaos as well.NEWLINENEWLINEThe paper is divided into four sections. The introduction gives an overview of the results motivating the problem addressed in this paper, an insightful outline of the main ideas and results, and also relevant definitions. In Section 2, the authors present the construction of the aforementioned family of Denjoy diffeomorphisms. This is based on Mañé's derived-from-Anosov diffeomorphisms. They prove the minimality of both the unstable and the central foliations. In Section 3, they define the holonomy map \(f\) on \(\mathbb{T}^2\) induced by the unstable foliation. This is a homeomorphism satisfying several topological properties, as listed in the Main Theorem. In Section 4, the authors show the differentiability of the unstable foliation through the \(C^r\) section theorem.