On the Iterates of Digit Maps

2016/09/12 by Chase, Zachary
#11A07 #11A63 (Primary) #11B05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1609.03263

Abstract

Given a base b, a "digit map" is a map f: ℤ^{≥ 0} → ℤ^{≥ 0} of the form f(∑_i=0ⁿ a_ibⁱ) = ∑_i=0ⁿ f_*(a_i), 0 ≤ a_i ≤ b-1 for each i, where f_* : \0,1,…, b-1\ → ℤ^{≥ 0} satisfies f_*(0) = 0 and f_*(1) = 1. It has been proven for b=10 and f_*(m) = m², and various generalizations thereof, that there are arbitrarily long sequences of consecutive positive integers that end up at 1 under repeated application of f. In this paper, we significantly generalize these results, providing a complete classification of digit maps for which, given any periodic point n, there are arbitrarily long sequences of consecutive positive integers that end up n.

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