On the dynamics on the SU(2)-character variety of a once-punctured torus
Carlos Matheus1
1 Centre de Mathématiques Laurent Schwartz, CNRS (UMR 7640), École Polytechnique, 91128 Palaiseau, (France)
Séminaire de théorie spectrale et géométrie, Volume 35 (2017-2019), pp. 109-119.

The natural SL(2,ℤ)-action on the SU(2)-character variety of a once-punctured torus respects the level sets of the function κ describing the values k∈[-2,2] of the traces of the matrices associated to a small loop around the puncture.

In 1998, R. Brown used Moser’s twisting theorem from KAM theory to show that no element of SL(2,ℤ) can act ergodically on every level set κ -1 (k). As it turns out, Brown’s original argument seems to be missing a detail, namely, there is no discussion of the twist condition in his application of Moser’s twisting theorem.

In 2002, H. Rüssmann improved Moser’s twisting theorem by establishing the stability of (Brjuno) elliptic fixed points of real-analytic area-preserving maps independently of twist conditions.

In this note, we observe that Brown’s argument can be completed by applying Rüssmann’s theorem instead of Moser’s twisting theorem.

Published online:
DOI: 10.5802/tsg.365
Carlos Matheus 1

1 Centre de Mathématiques Laurent Schwartz, CNRS (UMR 7640), École Polytechnique, 91128 Palaiseau, (France) 
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Carlos Matheus. On the dynamics on the $SU(2)$-character variety of a once-punctured torus. Séminaire de théorie spectrale et géométrie, Volume 35 (2017-2019), pp. 109-119. doi : 10.5802/tsg.365. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.365/

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