Some problems in arithmetic dynamics and related areas

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Dynamical systems are ubiquitous in modern mathematics. A discrete dynamical system is a set S together with a map f from S to itself, and dynamics is the study of the family of iterates {f, f2=f o f, f3=f o f o f,...}. For example, when S is the set of real numbers and f is the map that sends a number to its square, the family of iterates becomes {f(x)=x2, f2(x)=x4, f3(x)=x8,...}. I propose to study the situation when S is an object defined by polynomial equations (i.e. an algebraic variety) and the map f can be described by polynomials (i.e. algebraic morphism). This proposal has two parts.******The first part involves the principle of unlikely intersections suggesting that if the intersection of two objects is larger than expected, there should be an underlying geometric reason. As the simplest example, we have that if the intersection of two lines contains more than one point then the lines coincide. This somewhat naive principle is an important driving force behind significant developments in diophantine geometry and arithmetic dynamics in the last 10 years. Among many results on unlikely intersections in dynamics that I obtained, the most recent one is a new bounded height phenomenon in dynamics motivated by recent work of Amoroso, Masser, and Zannier. I propose to improve the techniques in my earlier work and discover new strategies to prove bounded height results for many more dynamical systems. In addition, I believe that the techniques used in the above work will have more applications to other problems in arithmetic dynamics and I also plan to study such applications.******The second part is about applications of results and ideas in arithmetic dynamics to related areas. This illustrates that arithmetic dynamics is not only interesting on its own but also has applications to highly fascinating problems in other areas. There are two "types" of applications that I wish to study: the first type involves the "arithmetic part" while the second type involves the "dynamics part" of arithmetic dynamics. More specifically, for the first type, I propose to prove algebraic independence of certain Mahler functions which played a significant role in transcendental number theory since the 1970s. My proposed research might help settle the notoriously difficult "non-vanishing step" in Mahler's theory for such functions. For the second type, I aim to discover new connections and analogies between arithmetic dynamics and other theories of dynamical systems. Arithmetic dynamics is a young area, and in a certain sense, descended directly from diophantine geometry and complex dynamics. On the other hand, there are several well established theories concerning dynamical systems such as topological dynamics, ergodic theory, symbolic dynamics (with applications to information theory), etc. Although this second aspect is less definite at the moment, it has the potential to open up highly interesting research directions in arithmetic dynamics.