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Centralizers of rational functions. (English) Zbl 1030.30032

Let \(R_d\) be a space of rational functions of degree \(d\) endowed with the \(C^0\) topology. The author studies the centralizer \(Z(f)\) of \(f\in R_d\), that is the set of rational functions which have degree greater than 1 and commute with \(f\) (in the sense of composition). \(f\) is said to have a trivial centralizer if \(Z(f)\) reduces to the iterates \(\{f^n\}\) of \(f\). The problem under investigation is whether the set of functions having trivial centralizer is massive enough. The main result of the paper claims that this set contains an open and dense set of rational functions.

MSC:

30D50 Blaschke products, etc. (MSC2000)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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