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Dynamical Systems from a Number Theorists Perspective

Joe Silverman
Brown University

Thursday, January 9, 2003
102 Bradley Hall, 4 pm
Tea 3:30 pm, Math Lounge

Abstract: A classical problem in the theory of dynamical systems is to describe the behavior of points under interation~$\phi^n=\phi\circ\phi\circ\cdots\circ\phi$ of a rational map~$\phi(z)=F(z)/G(z)$, i.e., where~$F(z)$ and~$G(z)$ are polynomials. The \textit{orbit} of a point~$\alpha$ under iteration of~$\phi$, denoted~$O_\alpha(\phi)$, is the set of images of~$\alpha$ under the iterates of~$\phi$, $O_\alpha(\phi)=\{\phi^n(\alpha):n\ge0\}$. The points with finite orbit, called \textit{preperiodic points}, play a particularly important role in the dynamics of~$\phi$. For a number theorist, it is natural to take~$F(z)$ and~$G(z)$ to have integer coefficients and to study the orbits of rational numbers~$\alpha\in{\Bbb Q}$. In this talk I will survey some of the known results and some of the outstanding conjectures related to this number-theoretic view of dynamics. Typical problems include: (1)~How many preperiodic points can be rational numbers~$\alpha\in{\Bbb Q}$? (2)~For which rational maps~$\phi$ can the orbit~$O_\alpha(\phi)$ of a rational number~$\alpha$ contain infinitely many integers?

This talk will be accessible to graduate students.