Generalized Collatz Maps with Almost Bounded Orbits

If dividing by p is a mistake, multiply by q and translate, and so you’ll live to iterate. The preceeding expresses a rule of form that motivates the generalized Collatz maps we study in this document. We show that if we define a Collatz-like map in this form then, under suitable conditions on p and q, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., p = 2 and q = 3). In other words, given an arbitrary growth function N → f (N) we show that almost every orbit of such map with input N eventually attains a value smaller than f (N).