{"paper":{"title":"Polynomial semiconjugacies, decompositions of iterations, and invariant curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.DS"],"primary_cat":"math.NT","authors_text":"Fedor Pakovich","submitted_at":"2015-05-23T16:53:28Z","abstract_excerpt":"We study the functional equation $A\\circ X=X\\circ B$, where $A,$ $B$, and $X$ are polynomials over $\\mathbb C$. Using previous results of the author about polynomials sharing preimages of compact sets, 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 about invariant curves of maps $F:\\,\\mathbb C^2 \\rightarrow \\mathbb C^2$ of the form $(x,y)\\rightarrow (f(x),f(y))$, where $f$ is a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06351","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}