$C^1$-actions of Baumslag-Solitar groups on $S^1$

TL;DR

This paper proves that any $C^1$ action of the Baumslag-Solitar group $BS(1,n)$ on the circle is essentially equivalent to its standard affine action, up to finite index and semiconjugacy.

Contribution

It establishes that all $C^1$ representations of $BS(1,n)$ on $S^1$ are semiconjugate to the standard affine action, revealing a rigidity property.

Findings

Any $C^1$ $BS(1,n)$ action on $S^1$ is semiconjugate to the standard affine action.

The result applies up to a finite index subgroup, indicating a form of structural stability.

The paper characterizes the dynamics of $BS(1,n)$ actions on the circle in the $C^1$ category.

Abstract

Let $BS(1, n)=< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that B(1, n) is isomorphic to the group generated by the two affine maps of the line : $f_0(x) = x + 1$ and $h_0(x) = nx $. The action on $S^1 = \RR \cup {\infty}$ generated by these two affine maps $f_0$ and $h_0 $ is called the standard affine one. We prove that any representation of BS(1,n) into $Diff^1(S^1)$ is (up to a finite index subgroup) semiconjugated to the standard affine action.