Grant: Discovery Grants Program - Individual

For any one-variable polynomial f with complex coefficients, and any complex number x, we call the f-orbit of x, the set consisting of x and all of its images under iterates of f. Complex dynamics studies properties of such orbits. Of particular interests are answers to questions such as: 1) When is the orbit living in a bounded set? 2) When is the orbit finite? For how many points x would this happen? 3) What can we tell about the common points for two such orbits corresponding to two different dynamical systems? Question 1) generated the definition of Julia and Fatou sets, whose topology was the main topic of study in complex dynamics for many years in the past. Question 2) is easy to answer over the field of complex numbers. There are infinitely many points whose orbit is finite, and they are called preperiodic. However, if we restrict our attention to the preperiodic points living in a finite extension of the field of rational numbers, then finding all preperiodic points becomes a much harder problem. Question 3) turned out to be much more elusive for complex dynamics, because it involves solving equations such as f^m(x) = g^n(y) (for fixed f, g, x and y), where the unknowns are non-negative integers n and m (note that f^m and g^n are the iterates of f and, respectively g). Furthermore, the above problem can be reformulated as follows: ``Given polynomials f and g with complex coefficients, and given complex numbers x and y, are there infinitely many pairs of non-negative integers (m,n) such that the point (f^m(x),g^n(y)) belongs to the diagonal subvariety of the affine plane?'' Now this is a question in arithmetic geometry, and this is precisely one of the questions that my research addresses, as I am looking for answering number theoretical questions in the context of complex dynamics, with the use of arithmetic geometry.