• Corpus ID: 202719469

# Conservation of Singularities in Functional Equations Associated to Collatz-Type Dynamical Systems; or, Dreamcatchers for Hydra Maps

@article{Siegel2019ConservationOS,
title={Conservation of Singularities in Functional Equations Associated to Collatz-Type Dynamical Systems; or, Dreamcatchers for Hydra Maps},
author={Maxwell C. Siegel},
journal={arXiv: General Mathematics},
year={2019}
}
• M. C. Siegel
• Published 20 September 2019
• Mathematics
• arXiv: General Mathematics
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…
2 Citations
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
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

## References

SHOWING 1-10 OF 44 REFERENCES
Some Analytic Aspects Concerning the Collatz Problem
A series of relatively simple equivalences to the Collatz conjecture, concerning the Collatz mapping [...] are presented. [...] The main topic of this paper consists in investigating a certain linear
Classical Topics in Complex Function Theory
An ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician.
ON A THEOREM OF SZEGO
is finite. Assume that the analytic function u(r, x) can be continued across some arc of the boundary of the unit circle. Then the ak are equal, beyond some point, to the terms of a periodic
A Paley-Wiener theorem for locally compact Abelian groups
Extending the Paley-Wiener theorem to locally compact Abelian groups involves both finding a suitable Laplace transform and a suitable analogue for analytic functions. The Laplace transform is
The Pontryagin duals of Q / Z and Q
Write S = {z ∈ C : |z| = 1}. A character of a locally compact abelian group G is a continuous group homomorphism G → S. We denote by Ĝ the set of characters of G, where for φ1, φ2 ∈ Ĝ and x ∈ G, we
The general theory of Dirichlet's series
V The Fundamental Properties of Analyt'ic Functions ; Taylor's, Latirent's, and Liouville's Theorems ; VI. The Theory of Residues ; Application to the Evaluation of Definite Integrals ; VIL. The
A course in abstract harmonic analysis
Banach Algebras and Spectral Theory Banach Algebras: Basic Concepts Gelfand Theory Nonunital Banach Algebras The Spectral Theorem Spectral Theory of *-Representations Von Neumann Algebras Notes and
Singularity Analysis of Generating Functions
• Mathematics
SIAM J. Discret. Math.
• 1990
This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic
Mellin Transforms and Asymptotics: Harmonic Sums
• Mathematics
Theor. Comput. Sci.
• 1995