Proof of a positivity conjecture of M. Kontsevich on non-commutative   cluster variables
  We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients. 
Kyungyong Lee and Ralf Schiffler
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