A short proof of Kontsevich’s cluster conjecture. (English) Zbl 1266.16026

Berenstein, Arkady; Retakh, Vladimir

C. R., Math., Acad. Sci. Paris 349, No. 3-4, 119-122 (2011).

Summary: We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map \(K_r\colon(x,y)\mapsto(xyx^{-1},(1+y^r)x^{-1})\) are given by noncommutative Laurent polynomials.

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

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