On the Iterates of Digit Maps
Zachary Chase

TL;DR
This paper classifies digit maps based on their iterative behavior, showing that for any periodic point, there exist arbitrarily long sequences of consecutive integers converging to it under iteration.
Contribution
It provides a complete classification of digit maps regarding the existence of arbitrarily long sequences converging to any given periodic point.
Findings
Classified digit maps with respect to their iterative convergence properties.
Extended previous results from specific cases like base 10 and quadratic functions.
Established conditions for the existence of long sequences ending at any periodic point.
Abstract
Given a base , a "digit map" is a map of the form , for each , where satisfies and . It has been proven for and , and various generalizations thereof, that there are arbitrarily long sequences of consecutive positive integers that end up at under repeated application of . In this paper, we significantly generalize these results, providing a complete classification of digit maps for which, given any periodic point , there are arbitrarily long sequences of consecutive positive integers that end up .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · Coding theory and cryptography · Mathematical Dynamics and Fractals
