# Dynamics of the $a$-map over residually finite Dedekind Domains and applications

## TL;DR

This paper investigates the behavior of the multiplication map over quotient rings of residually finite Dedekind domains, providing a detailed description of its dynamics and exploring various applications.

## Contribution

It offers a novel analysis of the dynamics of the $a$-map over these algebraic structures, extending previous work to a broader class of rings.

## Key findings

- Characterization of the $a$-map dynamics over quotient rings
- Identification of structural properties influencing the map's behavior
- Applications demonstrating the utility of the main results

## Abstract

Let $\mathfrak D$ be a residually finite Dedekind domain, $a\in \mathfrak D$ be a nonzero element and $\mathfrak n$ be a nonzero ideal of $\mathfrak D$. In this paper we describe the dynamics of the map $x\mapsto ax$ over the quotient ring $\mathfrak D/\mathfrak n$. We further present some applications of our main result.