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Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables. (English) Zbl 1266.16027
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.

##### 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
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##### 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
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