Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A $p$-Adic Characterization of the Periodic Points of a Class of
  Collatz-Type Maps on the Integers   Fix an integer $\varrho\geq2$. We consider maps
$H:\mathbb{Z}\rightarrow\mathbb{Z}$ so that for any $n\in\mathbb{Z}$, the value
of $H\left(n\right)$ is the image of one of \varrho distinct affine-linear
maps, the choice of which is determined by the value of $n$ mod $\varrho$;
these maps generalize the infamous Collatz or $3x+1$ map. This paper introduces
a $p$-adic formalism for studying the sets of periodic points of $H$,
applicable to the case where $H$ is the Collatz map, as well as a large family
of generalizations thereof. When applicable, this approach yields a function
$\chi_{H}:\mathbb{Z}_{\varrho}\rightarrow\mathbb{Z}_{q}$ (where $q$ is an
integer $\geq2$ associated to $H$ and where $q$ is an integer depending on $H$)
with the property that every element of
$\chi_{H}\left(\mathbb{Z}_{\varrho}\right)\cap\mathbb{Z}$ is a periodic point
of $\chi_{H}$, and that every cycle of $H$ of length $\geq2$ contains an
element of $\chi_{H}\left(\mathbb{Z}_{\varrho}\right)\cap\mathbb{Z}$. The
$\chi_{H}$ formalism simplifies, unifies, and generalizes previous work by
other authors on Collatz-type maps (both old and recent), and establishes a
foundation for future work on the periodic points of Collatz-type maps using
methods of $p$-adic Fourier analysis, Complex analysis, and Probability theory.

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The categories of this paper are: q-fin
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ