On Sarnak's conjecture and Veech's question for interval exchanges
type: Item
instance of: scholarly article
title: On Sarnak's conjecture and Veech's question for interval exchanges
zbMATH Open document ID: 1411.37008
author: Sébastien Ferenczi
author: Christian Mauduit
published in: Journal d'Analyse Mathématique
publication date: 2018-10-02T00:00:00Z
review text: 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…
reviewed by: Liang-Chung Hsia
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 6946526
zbMATH Keywords: interval exchange maps
zbMATH Keywords: weakly mixing
zbMATH Keywords: Rauzy induction
zbMATH Keywords: symbolic dynamics
zbMATH Keywords: Rokhlin towers
zbMATH Keywords: Sarnak's conjecture
zbMATH Keywords: Möbius function
Wikidata QID: Q123008644
MaRDI profile type: Publication
full work available at URL: s11854-018-0017-z
OpenAlex ID: W2772011410
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DOI: 10.1007/S11854-018-0017-Z
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