Topics in Arithmetic Dynamics

Abstract

This thesis comprises four papers completed during my doctoral studies at the University of Waterloo and is organized into three chapters.

The first chapter concerns preimage problems and dynamical cancellation, incorporating the papers “Dynamical Cancellation of Polynomials,” published in the Bulletin of the London Mathematical Society, and “Preimages Question for Surjective Endomorphisms on (P1)^n,” published in the New York Journal of Mathematics. We investigate stabilization phenomena for the set of rational points occurring in the preimages of invariant subvarieties under algebraic dynamical systems. In the special case in which the subvariety is the diagonal subvariety in P1 × P1 and the dynamics is given by a pair of rational functions (f, f), the problem reduces to a dynamical cancellation question. We also obtain a generalization of dynamical cancellation to the setting in which the dynamics are generated by a semigroup of polynomials.

The second chapter contains the joint paper with Chatchai Noytaptim, “A Finiteness Result for Common Zeros of Iterates of Rational Functions,” published in International Mathematics Research Notices. Addressing a question posed by Hsia and Tucker concerning finiteness properties of greatest common divisors of polynomial iterates, we prove that if f, g belong to C(X) and are compositionally independent rational functions, and c belongs to C(X), then, apart from a few explicit exceptional families with f and g in Aut(P1_C), there exist only finitely many λ in C for which there is an n satisfying f^n(λ) = g^n(λ) = c(λ).

The final chapter presents my recent preprint, “Polynomial Endomorphisms of A2 with Many Periodic Curves.” We show that for any regular polynomial endomorphism of positive degree on P2, every family of curves containing a Zariski dense set of periodic curves must be invariant under some iterate of the map. This establishes a weaker form of the Relative Dynamical Manin-Mumford Conjecture of DeMarco and Mavraki in the setting where the endomorphism is fixed in the family of dynamical systems, and may also be viewed as a dynamical Manin-Mumford statement on the moduli space of divisors. As an application, we classify all regular polynomial endomorphisms of P2 that admit infinitely many periodic curves of bounded degree.