Iterating sine, equivalence classes of variable changes, and groups with few conjugacy classes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916802133491712 |
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| author | Etingof, Pavel |
| author_facet | Etingof, Pavel |
| contents | This is an expository paper about iterations of a smooth real function $f$ on $[0,\varepsilon)$ such that $f(0)=0$, $f'(0)=1$, and $f(x)<x$ for $x>0$, 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 $f(x)=x+a_2x^2+a_3x^2+...$ with real coefficients (under composition). The same classification applies over a finite field $\mathbb{F}_p$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iterating sine, equivalence classes of variable changes, and groups with few conjugacy classes Etingof, Pavel History and Overview This is an expository paper about iterations of a smooth real function $f$ on $[0,\varepsilon)$ such that $f(0)=0$, $f'(0)=1$, and $f(x)<x$ for $x>0$, 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 $f(x)=x+a_2x^2+a_3x^2+...$ with real coefficients (under composition). The same classification applies over a finite field $\mathbb{F}_p$ 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. |
| title | Iterating sine, equivalence classes of variable changes, and groups with few conjugacy classes |
| topic | History and Overview |
| url | https://arxiv.org/abs/2506.16364 |