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Backward stability for polynomial maps with locally connected Julia sets

Authors:	Alexander Blokh Lex Oversteegen

Institution:	Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170 ; Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

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 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 depending on .

Keywords:	Complex dynamics locally connected Julia set backward stability conformal measure

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