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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Backward stability for polynomial maps with locally connected Julia sets
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by Alexander Blokh and Lex Oversteegen PDF
Trans. Amer. Math. Soc. 356 (2004), 119-133 Request permission

Abstract:

We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial $f$ with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure $\mu$ this easily implies that one of the following holds: 1. for $\mu$-a.e. $x\in J(f)$, $\omega (x)=J(f)$; 2. for $\mu$-a.e. $x\in J(f)$, $\omega (x)=\omega (c(x))$ for a critical point $c(x)$ depending on $x$.
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Additional Information
  • Alexander Blokh
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
  • MR Author ID: 196866
  • Email: ablokh@math.uab.edu
  • Lex Oversteegen
  • Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
  • MR Author ID: 134850
  • Email: overstee@math.uab.edu
  • Received by editor(s): October 10, 2001
  • Published electronically: August 25, 2003
  • Additional Notes: The first author was partially supported by NSF Grant DMS-9970363 and the second author by NSF grant DMS-0072626
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 119-133
  • MSC (2000): Primary 37F10; Secondary 37E25
  • DOI: https://doi.org/10.1090/S0002-9947-03-03415-9
  • MathSciNet review: 2020026