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Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo

"Recurrence in Algebraic Dynamics"

Let $\varphi:X\dashrightarrow X$ is a rational mapping of an algebraic variety $X$ defined over $\C$. The orbit of a point $x\in X$ is the sequence $\{x,\varphi(x),\varphi^2(x),\ldots\}$. Our basic question is: how often does this orbit intersect a given closed set $C$? Thus we are interested in the return set

\[ E := \{n\geq 0 : \varphi^n(x)\in C\}. \]