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Functional Equations Associated to Collatz-Type Maps on Integer Rings of Algebraic Number Fields
In 1995, Meinardus & Berg presented a reformulation of the Collatz Conjecture in terms of a functional equation in a single complex variable over the open unit disk. This paper generalizes that…
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Fourier Analysis of the Parity-Vector Parameterization of the Generalized Collatz px+1 maps
Let p be an odd prime, and consider the map H_p which sends an integer x to either x/2 or (px+1)/2 depending on whether x is even or odd. The values at x=0 of arbitrary composition sequences of the…
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Syracuse Random Variables and the Periodic Points of Collatz-type maps
Let $p$ be an odd prime, and consider the map $H_{p}$ which sends an integer $x$ to either $\frac{x}{2}$ or $\frac{px+1}{2}$ depending on whether $x$ is even or odd. The values at $x=0$ of arbitrary…
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Conservation of Singularities in Functional Equations Associated to Collatz-Type Dynamical Systems; or, Dreamcatchers for Hydra Maps
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…
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A $p$-Adic Characterization of the Periodic Points of a Class of Collatz-Type Maps on the Integers
Fix an integer ̺ ≥ 2. We consider maps H : Z → Z so that for any n ∈ Z, the value of H (n) is the image of one of ̺ distinct affine-linear maps, the choice of which is determined by the value of n…
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