The Critical Orbit Structure of Quadratic Polynomials in Zp

In this thesis, we develop a non-Archimedean analog to the Hubbard tree, a well-understood object from classical dynamics studied over the complex numbers. To that end, we explore the critical orbit structure of quadratic polynomials f_c(z) = z^2 + c with parameters c in the ring of p-adic rational integers, Zp. All such polynomials are post-critically bounded (PCB), and some are post-critically finite (PCF), which means that the forward orbit of the critical point, 0, is finite. If the orbit of 0 is finite, there exist minimal integers m and n such that f_c^{m+n}(0)=f_c^m(0), and we call (m,n) the critical orbit type of f_c. All PCF polynomials f defined over the complex numbers have an associated Hubbard tree which illustrates the orbit type of the critical points, and the geometry of those orbits within the filled Julia set of f. In order to define a p-adic analog, we give a description of the exact critical orbit type for PCF polynomials f_c defined over Zp, and then use the proximity of PCB parameters to PCF points in order to prove statements about the structure of the infinite critical orbit as visualized in the Zp tree.