Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
"Recurrence in Algebraic Dynamics"
Let $\varphi:X\dashrightarrow X$ is a rational mapping of an algebraic variety $X$ defined over $\C$. The orbit of a point $x\in X$ is the sequence $\{x,\varphi(x),\varphi^2(x),\ldots\}$. Our basic question is: how often does this orbit intersect a given closed set $C$? Thus we are interested in the return set
\[ E := \{n\geq 0 : \varphi^n(x)\in C\}. \]
Is it possible for $E$ to be the set of primes? Or the set of perfect squares? The Dynamical Mordell-Lang Conjecture (DML) says no: it asserts that $E$ is infinite only when it contains an infinite arithmetic progression. Geometrically, if the orbit intersects $C$ infinitely often, then in fact this intersection must occur periodically.
Although the DML Conjecture remains open in general, an elegant approach of Bell-Ghioca-Tucker obtains this periodicity when $E$ is a set of positive density. In this thesis, our first result is the generalization of the Bell-Ghioca-Tucker Theorem to the action an amenable semigroup on an algebraic variety (these are the semigroups in which "density" can be naturally defined). We also use ultrafilters to provide a combinatorial version for arbitrary semigroups; as a simple example, our result shows that the set $E$ cannot be equal to the ternary automatic set $\{n\in \N : [n]_3 \text{ has no 2's}\}$. Second, in joint work with Bell and Chen, we investigate dynamical sequences of the form $u_n=f(\varphi^n(x))$, where $f:X\dashrightarrow K$ is a rational function; we obtain several DML-type conclusions for this sequence, consequently recovering classical combinatorial theorems of Bézivin, Methfessel, and Polyá. Third, an investigation of other types of noetherian algebraic objects leads us to polycyclic-by-finite groups, and we prove an analogue of the Bell-Ghioca-Tucker Theorem for an automorphism of such a group.
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