arXiv subjects

Maxwell Charles Siegel

Publications and source records attributed to Maxwell Charles Siegel.

3 recordsLinked to original sources

arXiv2024

$\left(p,q\right)$-adic Analysis and the Collatz Conjecture

What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. '$\left(p,q\right)$-adic analysis' of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map $H:\mathbb{Z}\rightarrow\mathbb{Z}$, one can construct a function $χ_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ for an appropriate choice of distinct primes $p,q$ with the property that $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point of $H$ if and only if there is a $p$-adic integer $\mathfrak{z}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,\ldots\right\} $ so that $χ_{H}\left(\mathfrak{z}\right)=x$. By generalizing Monna-Springer integration theory and establishing a $\left(p,q\right)$-adic analogue of the Wiener Tauberian Theorem, one can show that the question 'is $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ a periodic point of $H$?' is essentially equivalent to 'is the span of the translates of the Fourier transform of $χ_{H}\left(\mathfrak{z}\right)-x$ dense in an appropriate non-archimedean function space?' This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice $\mathbb{Z}^{d}$ for any $d\geq1$.

math.GM↗

arXiv2023

The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory

Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of $T_{3}$ being an alternative formulation of the famous Collatz Conjecture. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed $\left(p,q\right)$-adic analysis, the study of functions from the $p$-adics to the $q$-adics, where $p$ and $q$ are distinct primes. In this, the first paper, working with the $T_{q}$ maps as a toy model for the more general theory, for each odd prime $q$, we construct a function $χ_{q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}$ (the Numen of $T_{q}$) and prove the Correspondence Principle (CP): $x\in\mathbb{Z}\backslash\left\{ 0\right\}$ is a periodic point of $T_{q}$ if and only there is a $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\}$ so that $χ_{q}\left(\mathfrak{z}\right)=x$. Additionally, if $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\mathbb{Q}$ makes $χ_{q}\left(\mathfrak{z}\right)\in\mathbb{Z}$, then the iterates of $χ_{q}\left(\mathfrak{z}\right)$ under $T_{q}$ tend to $+\infty$ or $-\infty$.

math.DS↗

arXiv2019

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 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.

math.GM↗