Backward stability for polynomial maps with locally connected Julia sets

Blokh, Alexander; Oversteegen, Lex

doi:10.1090/S0002-9947-03-03415-9

Backward stability for polynomial maps with locally connected Julia sets
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by Alexander Blokh and Lex Oversteegen

Trans. Amer. Math. Soc. 356 (2004), 119-133

DOI: https://doi.org/10.1090/S0002-9947-03-03415-9

Published electronically: August 25, 2003

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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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