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Mathematics Colloquium


Date:  Tuesday, February 19, 2013

Title:  The dynamical complexity of rational maps and an arithmetic analogue

Abstract:  Consider a rational map F = (F_1,...,F_N) consisting of an N-tuple of rational functions in N variables. The iterates F, F^2, F^3, ... of F determine a dynamical system whose complexity may be measured by the growth of the degree of F^n. A fundamental, and still quite mysterious, invariant is the dynamical degree D(F) of F, which is defined to be the limiting value of deg(F^n)^{1/n} as n goes to infinity. Recently people have also considered an arithmetic analogue of the dynamical degree in which one looks at the orbit of a point P having rational coordinates and replaces deg(F^n) by the arithmetic size of the coordinates of F^n(P). In this talk I will discuss dynamical degrees, arithmetic degrees, and various results and open problems that relate them. No background in dynamics, algebraic geometry, or number theory will be required.

Speaker:  Joe Silverman
Institution:  Brown University


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