Iterating Sine, Equivalence Classes of Variable Changes, and Groups with Few Conjugacy Classes
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283_2025_Article_10415.pdf
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1.17 MB
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c50ce98dbb625fea2e58d57c3b451dd5
Author(s)
Etingof, Pavel
Date Issued
July 23, 2025
Journal
The Mathematical Intelligencer
Publisher
Springer US
Citation
Etingof, P. Iterating Sine, Equivalence Classes of Variable Changes, and Groups with Few Conjugacy Classes. Math Intelligencer (2025).
Version
Final published version
Abstract
This is an expository paper about iterations of a
smooth real function f on [0, ) such that f(0) = 0, f E (0) = 1, and f(x) < x for x > 0, i.e., the sequence defined by xn+1 = f(xn). This sequence has interesting asymptotics, whose study leads to the question of classifying conjugacy classes in the group of formal changes of variable y = f(x), i.e., formal series f(x) = x + a2x2 + a3x2 + ⋯ with real coefficients (under composition). The same classification applies over a finite field p for suitably truncated series f, defining a family of p-groups that have the smallest number of conjugacy classes for a given order, i.e., are the “most noncommutative” finite groups currently known. The paper should be accessible to undergraduates and at least partially to advanced high school students.
smooth real function f on [0, ) such that f(0) = 0, f E (0) = 1, and f(x) < x for x > 0, i.e., the sequence defined by xn+1 = f(xn). This sequence has interesting asymptotics, whose study leads to the question of classifying conjugacy classes in the group of formal changes of variable y = f(x), i.e., formal series f(x) = x + a2x2 + a3x2 + ⋯ with real coefficients (under composition). The same classification applies over a finite field p for suitably truncated series f, defining a family of p-groups that have the smallest number of conjugacy classes for a given order, i.e., are the “most noncommutative” finite groups currently known. The paper should be accessible to undergraduates and at least partially to advanced high school students.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00283-025-10415-z
