## 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