Polynomial semiconjugacies, decompositions of iterations, and invariant\n curves

2015/05/23 by Fedor Pakovich, Pakovich, Fedor · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1505.06351

openalex publication_date 2015/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the functional equation A\∘ X=X\∘ B, where A, B, and X\nare polynomials over mathbb C. Using previous results of the author about\npolynomials sharing preimages of compact sets, we show that for given B its\nsolutions may be described in terms of the filled-in Julia set of B. On this\nbase, we prove a number of results describing a general structure of solutions.\nThe results obtained imply in particular the result of Medvedev and Scanlon\nabout invariant curves of maps F: , mathbb C² \→ mathbb C² of the\nform (x,y)\→ (f(x),f(y)), where f is a polynomial, and a version\nof the result of Zieve and M "uller about decompositions of iterations of a\npolynomial.\n

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