Polynomial semiconjugacies, decompositions of iterations, and invariant curves

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Abstract

We study the functional equation A o X = X o B, where A, B, and X are polynomials with complex coefficients. Using results of [13] about polynomials sharing preimages of compact sets in C, we show that for given B its solutions may be described in terms of the filled-in Julia set of B. On this base, we prove a number of results describing a general structure of solutions. The results obtained imply in particular the result of Medvedev and Scanlon [10] about invariant curves of maps F: ℂ2 → ℂ2 of the form (x, y) → (f(x), f(y)), where f is a polynomial, and a version of the result of Zieve and Müller [22] about decompositions of iterations of a polynomial.

Original languageEnglish
Pages (from-to)1417-1446
Number of pages30
JournalAnnali della Scuola normale superiore di Pisa - Classe di scienze
Volume17
Issue number4
StatePublished - 1 Jan 2017

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)

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