Lee, Kyungyong; Schiffler, Ralf Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables. (English) Zbl 1266.16027 Compos. Math. 148, No. 6, 1821-1832 (2012). Summary: We prove a conjecture of Kontsevich, which asserts that the iterations of the non-commutative rational map \(F_r\colon(x,y)\to (xyx^{-1},(1+y^r)x^{-1})\) are given by non-commutative Laurent polynomials with non-negative integer coefficients. Cited in 4 ReviewsCited in 8 Documents MSC: 16S38 Rings arising from noncommutative algebraic geometry 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 13F60 Cluster algebras Keywords:Kontsevich cluster conjecture; iterations of rational maps; noncommutative Laurent polynomials; lattice paths PDF BibTeX XML Cite \textit{K. Lee} and \textit{R. Schiffler}, Compos. Math. 148, No. 6, 1821--1832 (2012; Zbl 1266.16027) Full Text: DOI arXiv References: [1] doi:10.1016/j.aim.2011.05.017 · Zbl 1252.37069 · doi:10.1016/j.aim.2011.05.017 [2] doi:10.1016/j.crma.2011.01.004 · Zbl 1266.16026 · doi:10.1016/j.crma.2011.01.004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.