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Any Baumslag-Solitar action on surfaces with a pseudo-Anosov element has a finite orbit. (English) Zbl 1442.37044

This paper deals with Baumslag-Solitar group actions on surfaces with a pseudo-Anosov element. Let \(n \in \mathbb{N}\), \(n \geq 2\). The solvable Baumslag-Solitar group \(\operatorname{BS}(1,n)\) is defined by \(\operatorname{BS}(1,n)=\langle a,b | aba^{-1}=b^n\rangle\). The group \(\operatorname{BS}(1,n)\) can be represented as the subgroup generated by the maps \(a(x)=x+1\) and \(b(x)= nx\). We denote by \(\langle f, h\rangle\) an action of \(\operatorname{BS}(1,n)\) on a surface, meaning that the homeomorphisms \(f\) and \(h\) satisfy \(h \circ f \circ h^{-1} =f^n.\) Let \(S\) be a closed connected oriented surface embedded in the three-dimensional Euclidean space \(\mathbb{R}^3\), endowed with the usual norm, and \(F\) a map on \(S\), i.e., \(F:S\rightarrow S\) which is \(C^1\) on a open \(W\) containing \(S\). The differential of \(F\) at \(x \in W\) is a map \(\mathcal{D}F(x)=\mathcal{D}_x F: T_xS \rightarrow \mathbb{R}^3\). The authors define \[ \mathcal{S} (F, W)=\sup \{ \|\mathcal{D}_x F\|, x \in W\},\] \[ V_{\epsilon} (F)=\{ x \in S: \|(F-Id)(x)\|+\|\mathcal{D}_x (F-Id)\|<\epsilon\},\] and denote by \(\Lambda \) an \(F\)-invariant compact set; \(F\) is \(C_{\Lambda}^1\) if \(F\) is \(C^1\) on a neighborhood \(W\) of \(\Lambda\) in \(S\). The two main results in the paper read as follows:
Theorem 1. Let \(f\) and \(h\) be two homeomorphisms, generating a faithful action of \(\operatorname{BS}(1,n)\) on \(S\), \(\Lambda\) be a minimal set of \(\langle f, g\rangle\) included in the \(\mathrm{Fix} (f)\) and \(f,h\) are \(C^1_{\Lambda}\).
1. Any point \(x \in \Lambda\) is an \(f\)-elliptic fixed point, i.e., there exists a positive integer \(N\) such that the eigenvalues of \(D_x f^N\)is 1 for any \(x \in \Lambda\).
2. For all \(\epsilon >0\), there exist \(\delta >0\) and a \(C^1_{\Lambda}\)-diffeomorphism \(f_{\epsilon} \in \langle f, g\rangle\) such that
(a) \(f_{\epsilon}\), \(h\) generate a faithful action of \(\operatorname{BS}(1,n)\) on \(S\) and \(\Lambda \subset \mathrm{Fix} (f_{\epsilon}) \subset \mathrm{Fix} (F^N).\)
(b) \(\| f_{\epsilon}-Id \|_{B_{\delta}(\Lambda)}\leq \epsilon, \) where \(B_{\delta}(\Lambda)\) is the union of balls of center in \(\Lambda\) and radius \(\delta\).
Theorem 2. Let \(f, h\) and \(\Lambda\) be as in Theorem 1. Let \(W\) be a neighborhood of \(\Lambda\) such that \(f\) and \(h\) are \(C^1\) on \(W\). Suppose that any point \(x \in \Lambda\) has an \(h\)-unstable manifold \(W^u(x)\); then either there exists \(N \in \mathbb{N}\) such that \(W^u(x) \subset \mathrm{Fix} (f^N)\) for all \(x \in \Lambda\); or \(\mathcal{S} (h, W) \geq n/C_S\), for some constant \(C_S \geq 1.\)
Several corollaries and other interesting results are also derived. Among them we have that any Baumslag-Solitar action on a surface with a pseudo-Anosov element has a finite orbit.

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
22F05 General theory of group and pseudogroup actions
22F10 Measurable group actions

Citations:

Zbl 1311.37014
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References:

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