Directions in arithmetic dynamics

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This project focuses on the subject of algebraic dynamics. Orbits of points under iterates of maps are among the most important objects in dynamics. In the study of algebraic dynamics, the map is typically a polynomial or rational map in one or more variables. One natural question is what kinds of algebraic relations the forward orbits of points can satisfy. When the orbits are infinite (corresponding to so-called "wandering points"), one natural question corresponds to the well-known Mordell-Lang conjecture for abelian varieties. When the orbits are finite (corresponding to "preperiodic points'), a natural question is an analog of the Manin-Mumford-Bogomolov conjecture for abelian varieties. The primary purpose of this project is to advance knowledge about these two questions, via a combination of techniques from p-adic analysis, geometry, and diophantine approximation.This project focuses on the interaction between algebraic maps and algebraic equations. An algebraic map is a function such as f(x) = 2x+3. Applying the map repeatedly to a single number gives what is called the orbit of that number under f. For example, if f(x) = 2x+3 and 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, and n-tuples of numbers by allowing f to be an algebraic map in more than one variable. An algebraic equation is simply a more general version of the familiar quadratic polynomials one encounters in beginning algebra, except that it may have more variables and have higher degree terms. These orbits exhibit many interesting properties. On the one hand they have so little structure that they may be good engines for the generation of random numbers; on the other, one might hope that their algebraic and arithmetic properties may give rise to new analogs the theories of fractals and chaos, which in many cases arose from the consideration of geometric and analytical properties of orbits.