Attractors and recurrence for dendrite-critical polynomials |
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Authors: | Alexander Blokh Micha? Misiurewicz |
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Affiliation: | a Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-2060, USA b Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA |
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Abstract: | We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point. |
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Keywords: | Complex dynamics Attractors Conformal measures Postcritical set |
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