Yuri Kozitsky and Lech Wolowski
A nonlinear dynamical system on the set 
of Laguerre entire functions 
(358K, PostScrip)

ABSTRACT.  A nonlinear modification of the Cauchy problem ${\partial} f 
(t, z)/{\partial t}= 
\theta D_z f(t,z) + zD^2_z f(t,z)$, $D_z = \partial /\partial z$, 
 $t\in {I\!\! R}_+ = [0, +\infty)$, 
 $z\in \hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5pt}C$, 
 $\theta \geq 0 $, $f(0,z) = g(z)$, $g\in 
{\mathcal L}$ is considered. The set ${\mathcal L}$ consists of 
Laguerre entire functions, which one obtains as a closure of the 
set of polynomials having real nonpositive zeros only in the 
topology of uniform convergence on compact subsets of $ \ 
\hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5pt}C$. 
The modification means that the time half-line ${I\!\! R}_+$ is 
divided into the intervals ${\mathcal I}_n = [(n-1)\tau , n\tau 
]$, $n\in {I\!\! N}$, $\tau>0$, and on each ${\mathcal I}_n $ the 
evolution is to be described by the above equation but at the 
endpoints the function $f(t, z)$ is changed: $f(n\tau , z) 
\rightarrow \left[ f\left(n\tau , 
z\delta^{-1-\lambda}\right)\right]^\delta $, with $\lambda>0$ and 
an integer $\delta \geq 2$. The resolving operator of such problem 
preserves the set ${\mathcal L}$. It is shown that for 
$t\rightarrow +\infty$, the asymptotic properties of $f(t, z)$ 
change considerably when the parameter $\tau$ reaches a threshold 
value $\tau_*$. The limit theorems for $\tau < \tau_* $ and for 
$\tau = \tau_* $ are proven. Applications, including limit 
theorems for weakly and strongly dependent random vectors, are 
given.