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\begin{document}

\title[Dynamics of quadratic polynomials: Complex Bounds]%
{Dynamics of quadratic polynomials:\\Complex bounds for real maps }
\author {Mikhail Lyubich }
\address{
Mikhail Lyubich\\
Mathematics Department\\
SUNY at Stony Brook\\
Stony Brook, NY 11794}
\email{
mlyubich@math.sunysb.edu}
\author { Michael Yampolsky}
\address{
Michael Yampolsky\\
Mathematics Department\\
SUNY at Stony Brook\\
Stony Brook, NY 11794}
\email {yampol@math.sunysb.edu}
\thanks{This work was supported in part by Sloan Research Fellowship
and by NSF grants DMS-8920768 and DMS-9022140 (at MSRI).}
\date{\today}

\maketitle

\begin{abstract}
We extend Sullivan's complex a priori bounds to real quadratic
polynomials with essentially bounded combinatorics. Combined with the
previous results of the first author, this yields complex bounds for
all real quadratics. Local connectivity of the corresponding Julia
sets follows.
\end{abstract}


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