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\topmatter
\title\nofrills Erratum:\\
On the spectra of\\
randomly perturbed expanding maps
\endtitle
\author V. Baladi and L.-S. Young
\endauthor
\address
Mathematik, ETH Z\"urich, CH 8092 Z\"urich, Switzerland
\hfill\break
(on leave from UMPA, ENS Lyon (CNRS, UMR 128), France)
\endaddress
\email
baladi\@math.ethz.ch
\endemail
\address
Department of Mathematics, UCLA, Los Angeles, CA 90024, USA
\endaddress
\email lsy\@math.ucla.edu \endemail
\date{June 1994}
\enddate
\endtopmatter
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\document
The authors wish to point out an error in Sublemma 6 in Section 5
of [1]. The claims in Theorems 3 and 3' have been revised accordingly;
a correct version is given below. Other results in [1] are
not affected.
The first author is grateful to P. Collet, S. Isola, and B. Schmitt
for useful discussions.
\subhead i) Revised statement of results
in Section 5.C
\endsubhead
Section 5 of [1] is about piecewise $C^2$ expanding
mixing maps $f$ of the interval. The number $\Theta$ below refers
to $\Theta = \lim_{n \to \infty} \sup (1/|(f^n)'|^{1/n})$.
These maps are randomly perturbed by taking convolution with
a kernel $\theta_\epsilon$, and the resulting Markov chain
is denoted $\XX^\epsilon$. The precise statements of Theorems 3
and 3' should read as follows:
\proclaim{Theorem 3}
Let $f: I\to I$ be
as described in Section 5.A of [1],
with a
unique absolutely continuous invariant
probability measure $\mu_0= \rho_0 \, dm$, and let $\XX^\epsilon$ be a
small random
perturbation of $f$ of the type described in Section 5.B
with invariant probability measure $\rho_\epsilon \, dm$.
We assume also that $f$ has no periodic turning points. Then
\roster
\item
The dynamical system $(f,\mu_0)$ is stochastically stable under
$\XX^\epsilon$ in $L^1(dm)$, i.e.,
$|\rho_\epsilon - \rho_0|_1$ tends to $0$ as
$\epsilon \to 0$.
\endroster
Let $\tau_0<1$ and $\tau_\epsilon<1$ be the rates of decay of correlations
for $f$ and $\XX^\epsilon$ respectively for
test functions in $BV$. Then:
\roster
\item[2]
$\limsup_{\epsilon \to 0} \tau_\epsilon \le \sqrt \tau_0$.
\endroster
\endproclaim
\proclaim{Theorem 3'}
Let $f$ and $\XX^\epsilon$ be as in Theorem 3, except that
we do not require that $f$ has no periodic turning points. Then
\roster
\item
$|\rho_\epsilon - \rho_0|_1$ tends to $0$ as
$\epsilon \to 0$ if $2 < 1/\tau_0 \le 1/\Theta$;
\item
$\limsup_{\epsilon \to 0} \tau_\epsilon \le \sqrt { 2\tau_0}$.
\endroster
If
$\theta_\epsilon$ is symmetric, the factor ``$2$'' in both
\therosteritem{1} and \therosteritem{2} may be
replaced by ``$3/2$''.
\endproclaim
\smallskip
Section 5.D is unchanged.
\smallskip
\subhead ii) Revised version of Section 5.E
\endsubhead
\smallskip
We follow the notation introduced at the beginning of 5.E, except
that we consider only the situation where
$$
\Sigma_0=\{1\} \quad \text{and} \quad \Sigma_{1,0} = \emptyset \, .
$$
That is to say, the reader should read 5.E with $\kappa_0=1$,
$\kappa_{11}=\kappa_1=\tau_0$, etc.
Sublemma 6, which is problematic in [1], is valid in this
more limited setting because
$\pi_0 \varphi = \rho_0 \cdot \int \varphi \, dm$.
Lemmas 1' and 3', which use Sublemma 6, are also correct under the
present assumptions. We take this opportunity to add
``$X^\epsilon_0 \to X_0$'', which had been inadvertently left out in [1],
to the conclusion of Lemma 3'.
To prove Theorem 3, one applies Lemmas 9, 1' and 3'
with $\kappa$ close to (and slightly bigger than)
$\sqrt \tau_0$. To prove Theorem 3', take $\kappa$ close to
$\sqrt{\tau_0/2}$ (or $\sqrt{\tau_0/(3/2)}$ if $\theta_\epsilon$
is symmetric).
\bigskip
\Refs
\ref \no 1
\by V. Baladi and L.-S. Young
\paper On the spectra of randomly
perturbed expanding maps
\jour Comm. Math. Phys.
\vol 156
\pages 355-385
\yr 1993
\endref
\endRefs
\enddocument