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

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

Published electronically:
August 25, 2003

MathSciNet review:
2020026

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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