Abstract: The theory of complex dynamical systems was initiated about a century ago, and interest was renewed in the 1980's by work of Douady, Hubbard, and others. Noting analogies between dynamical systems and various objects in algebraic geometry and number theory, Morton and Silverman began to develop an arithmetic theory of dynamical systems in the mid-1990's. I will discuss preperiodic points for polynomial maps, motivated by the problem of counting the number of such points in both the complex and arithmetic settings, and I will survey various results on questions of this type.
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