Iterating sine, equivalence classes of variable changes, and groups with few conjugacy classes

2025/06/19 by Etingof, Pavel
Computer Science · Mathematics · #FOS: Mathematics #Functional Equations Stability Results #History and Overview (math.HO) #Mathematical and Theoretical Analysis #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2506.16364

openalex publication_date 2025/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is an expository paper about iterations of a smooth real function f on [0,ε) such that f(0)=0, f'(0)=1, and f(x)0, i.e., the sequence defined by x_n+1=f(x_n). 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 \mathbbF_p for suitably truncated series f, defining a family of p-groups which 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.

Citations

• On residues and conjugacies for germs of 1-D parabolic diffeomorphisms in finite regularity
• Finite groups have more conjugacy classes
• Finite groups have even more conjugacy classes

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• Counting conjugacy classes in groups with contracting elements
• Conjugacy growth series of some infinitely generated groups
• Short character values on large conjugacy classes
• Twisted conjugacy growth series of virtually abelian groups
• Conjugacy classes, characters and products of elements