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Backward stability for polynomial maps with locally connected Julia sets

Authors: Alexander Blokh and Lex Oversteegen
Journal: Trans. Amer. Math. Soc. 356 (2004), 119-133
MSC (2000): Primary 37F10; Secondary 37E25
Published electronically: August 25, 2003
MathSciNet review: 2020026
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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

Lex Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Keywords: Complex dynamics, locally connected, Julia set, backward stability, conformal measure
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
Article copyright: © Copyright 2003 American Mathematical Society

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