Arithmetic dynamics

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Arithmetic dynamics is a new field that is an amalgamation of two areas of mathematics, dynamical systems and number theory. The subject can be viewed as the transfer of previous results in the theory of Diophantine equations to the setting of discrete dynamical systems.

Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures.

The correspondence between Diophantine equations and Dynamical systems can be outlined as follows: rational and integral points on algebraic varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational functions.

[edit] References

• Joseph H. Silverman (2007). The Arithmetic of Dynamical Systems. Springer. ISBN 978-0-387-69903-5.