[CvGmt News] Re: Abstract again... (fwd)

Dynamical Systems Trimester dynsys at math.sns.it
Tue May 6 10:15:24 CEST 2003


SEMINARIO DI SISTEMI DINAMICI (olomorfi e dintorni)

Aula Riunioni del Centro di Ricerca Matematica Ennio De Giorgi
Piazza dei Cavalieri, Collegio Puteano

Giovedi' 8 maggio, ore 17.00 
(dopo il "COLLOQUIO DE GIORGI" di Pesin).

Prof. Nessim Sibony

Titolo:
Polynomial like mapping in several variables.
(Joint work with T.C.Dinh)

Abstract:
\begin{abstract}
We study the dynamics of polynomial-like mappings in several variables.
A special case of our results is the following theorem.
\\
\it
Let $f:U\longrightarrow V$ be a proper holomorphic map
from an open set $U\subset \subset V$ onto a Stein manifold $V$.
Assume $f$ is
of topological degree $d_t\geq 2$. Then there
is a probability measure $\mu$ supported on $\K:=\bigcap_{n\geq 0}f^{-n}(V)$
satisfying the following properties.
\begin{enumerate}
\item[{\rm 1.}] The measure $\mu$ is invariant, K-mixing, of maximal entropy
   $\log d_t$.
\item[{\rm 2.}] If $J$ is the Jacobian of $f$ with respect to a volume form
   $\Omega$ then $\int \log J \d \mu \geq
   \log d_t$.
\item[{\rm 3.}] For every probability measure $\nu$ on $V$ with no mass on
   pluripolar sets $d_t^{-n} (f^n)^* \nu \rightharpoonup \mu$.
\item[{\rm 4.}]
If the p.s.h. functions on $V$ are $\mu$-integrables ($\mu$ is
   PLB) then
\begin{enumerate}
\item[{\rm (a)}] The Lyapounov exponents for $\mu$ are strictly positive.
\item[{\rm (b)}] $\mu$ is exponentially mixing.
\item[{\rm (c)}] There is a proper analytic
subset $\E_0$ of $V$ such that $f^{-1}(\E_0)\subset \E_0$ and for
   $z\not\in\E$, $\mu^z_n:=d_t^{-n} (f^n)^*\delta_z \rightharpoonup
   \mu$ where $\E=\cup_{n\geq 0} f^n(\E_0)$.
\item[{\rm (d)}] The measure $\mu$ is a limit of Dirac masses on the repelling
   periodic points.
\end{enumerate}
\end{enumerate}
The condition $\mu$ is PLB is stable under small pertubation of
$f$. This gives large families where it is satisfied.
\rm
\end{abstract}

Marco Abate
Stefano Marmi



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