Algebraic dynamics

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This project focuses on the subject of algebraic dynamics. Orbits ofpoints under iterates of maps are among the most important objects indynamics. In the study of algebraic dynamics, the map is typically apolynomial or rational map in one or more variables. It is natural toask what kinds of algebraic relations infinite families of elementswithin an orbit can satisfy. The PI conjectures that if infinitelymany elements within an orbit satisfy an algebraic relation, then thatrelation must in some sense be stable under some iterate of the map.This conjecture may be thought of as a general dynamical analog of thewell-known Mordell-Lang conjecture for group varieties. Thus far,this conjecture has been proved in a few special cases using acombination of diophantine and p-adic analytic methods. The main goalof this project is to prove the conjecture in general. This project focuses on the interaction between algebraic maps andalgebraic equations. An algebraic map is a function such as f(x) =2x+3. Applying the map repeatedly to a single number gives what iscalled the orbit of that number under f. For example, if f(x) = 2x+3and we start with the number 1, then the orbit is 1, 5, 13, 29, 61,and so on. One can also form orbits out of pairs, triplets, andn-tuples of numbers by allowing f to be an algebraic map in more thanone variable. An algebraic equation is simply a more general versionof the familiar quadratic polynomials one encounters in beginningalgebra, except that it may have more variables and have higher degreeterms. One expects that if infinitely many numbers in an orbitsatisfy some algebraic relation, then that relation has an innateconnection with the algebraic map. The PI states a precise conjecturealong these lines, which he hopes to prove. The strategy for theproof involves a wide range of techniques, some number theoretic, somegeometric, and some analytic.