  The Collatz Conjecture's connection to dynamical systems opens it to a
variety of techniques aimed at recurrence and density results. First, we turn
to density results and strengthen the result of Terras through finding a strict
rate of convergence. This rate gives a preliminary result on the Triangle
Conjecture, which describes a set nodes that would dominate $L^{C} = \{y \in \N
\, | \, T^{k}(y) > y , \,\forall k \in \N \}$. Second, we extend prior
arguments to show that the construction of several classes of measures imply
the bounded trajectories piece of the Collatz Conjecture.
