Rational functions admitting double decomposition

Abstract: J.Ritt has investigated the structure of complex polynomials with respect to superposition. In particular, he listed all the polynomials admitting different double decompositions into indecomposable polynomials. The analogues of Ritt theory for rational functions were constructed just for several particular classes of the said functions, say for Laurent polynomials (F.Pakovich). In this note we describe a certain class of double decompositions for rational functions. Essentially, described below rational functions were discovered by E.I.Zolotarev in 1877 as a solution of certain optimization problem. However, the double decomposition property for them was hidden until recently because of somewhat awkward representation. We give a (possibly new) symmetric representation of Zolotarev fractions resembling the parametric representation for Chebyshev polynomials, which are a special limit case of Zolotarev fraction.