Gregory Kelsey

Let $q$ be a prime power and $\phi$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $\phi$. Denote by $a_n$ the proportion of periodic elements. Little is known about how $a_n$ changes as $n$ grows, unless $\phi$ is a power map or Chebyshev polynomial. We give the first results on this question for a wider class of rational functions: $a_n$ has lim inf $0$ when $q$ is odd and $\phi$ is quadratic and neither Latt\`es nor conjugate to a one-parameter family of exceptional maps. We also show that $a_n$ has limit $0$ when $\phi$ is a non-Chebyshev quadratic polynomial with strictly preperiodic finite critical point and $q$ is an odd square. Our methods yield additional results on periodic points for reductions of post-critically finite (PCF) rational functions defined over number fields. The difficulty of understanding $a_n$ in general is that $\mathbb{P}^1(\F_{q^n})$ is a finite set with no ambient geometry. In fact, $\phi$ can be lifted to a PCF rational map on the Riemann sphere, where we show that $a_n$ is given by counting elements of the iterated monodromy group (IMG) that act with fixed points at all levels of the tree of preimages. Using a martingale convergence theorem, we translate the problem to determining whether certain IMG elements exist. This in turn can be decisively addressed using the expansion of PCF rational maps in the orbifold metric.

Given a bordified space, Karlsson defines an incidence geometry of stars at infinity. These stars and their incidence are closely related to well-understood objects when the space is hyperbolic, CAT(0), or a bounded convex domain with the Hilbert metric. A question stemming from Karlsson's original paper was whether or not the relation of one boundary point being included in a star of another boundary point is symmetric. This paper provides an example demonstrating that this relation in the star boundary of the three-tree Diestel-Leader graph $DL_3(q)$ is not symmetric. In doing so, some interesting bounds on distance in Diestel-Leader graphs are utilized.

We fully describe the horofunction boundary $\partial_h L_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary $\partial_\infty L_2$ with this metric is a subset of $\partial_h L_2$. Although $\partial_\infty L_2$ does not embed continuously in $\partial_h L_2$, it naturally splits into two subspaces, each of which is a punctured Cantor set and does embed continuously. The height function on $\text{DL}(2,2)$ provides a natural stratification of $\partial_h L_2$, in which countably-many non-Busemann points interpolate between the two halves of $\partial_\infty L_2$. Furthermore, the height function and its negation are themselves non-Busemann horofunctions in $\partial_h L_2$ and are global fixed points of the action of $L_2$.