On Sarnak's conjecture and Veech's question for interval exchanges
It is shown by \textit{A. Avila} and \textit{G. Forni} [Ann. Math. (2) 165, No. 2, 637--664 (2007; Zbl 1136.37003)] that almost all interval exchange maps are (measure-theoretically) weakly mixing. However, explicit examples (as opposed to existence theorems) of interval exchanges of weakly mixing are rare. A more difficult measure theoretical question asked by \textit{W. A. Veech} [Monatsh. Math. 94, 335--341 (1982; Zbl 0499.28016)] is whether or not almost every interval exchange maps is simple. Explicit examples of simple $k$-interval exchanges are only known for small values of $k$. We refer the readers to this paper under review for a brief account and related references. \par The main results of this paper give explicit constructions (i.e, with an explicit algorithm) of $k$-interval exchange maps that are weakly mixing in every Rauzy class as well as interval exchange maps that are simple. The authors prove that there are interval exchange maps which give a positive answer to Veech's question for more than four intervals and outside the hyperelliptic Rauzy class. \textit{P. Sarnak} [\url{https://publications.ias.edu/Sarnak/papers/512} (2011)] posed a conjecture asserting that any bounded sequence of complex numbers produced by a zero topological entropy (which include interval exchange maps) is (asymptotically) orthogonal to the Möbius function. The authors are able to show that Sarnak's conjecture holds for a large subfamilies of interval exchanges constructed in this paper. The methods for the constructions are mainly adapted from [\textit{V. Delecroix} and \textit{C. Ulcigrai}, Geom. Dedicata 176, 117--174 (2015; Zbl 1326.37023)] and the algorithms developed in [\textit{S. Ferenczi}, Ann. Inst. Fourier 64, No. 5, 1947--2002 (2014; Zbl 1352.37035)]. The main tools used in the construction are Rokhlin towers used for an explicit representation of the systems and the self-dual induction and its generalization. \par The authors extend a criterion due to \textit{J. Bourgain} [J. Anal. Math. 120, 105--130 (2013; Zbl 1358.37017)] under which Sarnak's conjecture holds. The criterion can be applied to a large family of constructions of $k$-interval exchanges that are built from selfdual induction and its generalization for all $k.$ For each primitive permutation, the authors construct examples of $k$-interval exchanges satisfying Sarnak's conjecture, and, for at least one permutation (called the standard permutation) in every Rauzy class with weak mixing and simplicity. Two different families of examples which are uniquely ergodic and satisfy Sarnak's conjecture using techniques of finite rank in one case, rank one in the other are given.
scientific article; zbMATH DE number 6946526
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