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 $\chi_{ H}:\mathbb{ Z}_{ p}\rightarrow\mathbb{ Z}_{ q}$ 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 $\chi_{ H}\left(\mathfrak{ z}\right)=x$. 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 $\chi_{ H}\left(\mathfrak{ z}\right)-x$ 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 $d\geq1$.
Categories: math.GM