Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $x$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37F10, 37E25

Retrieve articles in all journals with MSC (2000): 37F10, 37E25


Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: ablokh@math.uab.edu

Lex Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: overstee@math.uab.edu

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