| Popis: |
It is known that the Collatz Conjecture (and the study of similar maps, here called "Hydra maps") can be stated in terms of solution sets of functional equations; or, equivalently, the fixed points of linear operators on spaces of analytic functions. Rather than studying potential fixed points of such operators, we examine their effect on the singularities of functions. To that end, we introduce the notion of a "dreamcatcher", an object which encodes the location and "virtual residue" of the singularities of an analytic function of a specified growth rate along the boundary of the function's region of convergence. Dreamcatchers can be given rigorous footing as elements of the Hilbert space L2(Q/Z) of complex-valued functions on Q/Z which are square-integrable with respect to the counting measure on Q/Z. The aforementioned linear operators (called here "permutation operators") are shown to "conserve" the singularities of their fixed points, in the sense that the dreamcatcher of a fixed point is itself the fixed point of the operator induced on L2(Q/Z) (the "dreamcatcher operator") by the permutation operator. Using Pontryagin duality and p-adic analysis, the fixed points of the operators acting on dreamcatchers for simple-pole-like singularities are completely determined for a large class of Hydra maps in the case where the fixed points are finitely supported on Q/Z. This enables qualitative conclusions to be made about those Hydra maps' orbit classes in the non-negative integers. |
