Iterating Sine, Equivalence Classes of Variable Changes, and Groups with Few Conjugacy Classes

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.