The natural -action on the -character variety of a once-punctured torus respects the level sets of the function describing the values 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 can act ergodically on every level set . 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.
@article{TSG_2017-2019__35__109_0,
author = {Matheus, Carlos},
title = {On the dynamics on the $SU(2)$-character variety of a once-punctured torus},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
pages = {109--119},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {35},
year = {2017-2019},
doi = {10.5802/tsg.365},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/tsg.365/}
}
TY - JOUR AU - Matheus, Carlos TI - On the dynamics on the $SU(2)$-character variety of a once-punctured torus JO - Séminaire de théorie spectrale et géométrie PY - 2017-2019 SP - 109 EP - 119 VL - 35 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/tsg.365/ DO - 10.5802/tsg.365 LA - en ID - TSG_2017-2019__35__109_0 ER -
%0 Journal Article %A Matheus, Carlos %T On the dynamics on the $SU(2)$-character variety of a once-punctured torus %J Séminaire de théorie spectrale et géométrie %D 2017-2019 %P 109-119 %V 35 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/tsg.365/ %R 10.5802/tsg.365 %G en %F TSG_2017-2019__35__109_0
Matheus, Carlos. On the dynamics on the $SU(2)$-character variety of a once-punctured torus. Séminaire de théorie spectrale et géométrie, Tome 35 (2017-2019), pp. 109-119. doi : 10.5802/tsg.365. http://archive.numdam.org/articles/10.5802/tsg.365/
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