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We state a theorem relating the ergodicity of the action of a given subgroupof the mapping class group of a surface on the character variety, to theasymptotic of its invariant subspaces through the Witten-Reshetikhin-Turaevrepresentations. As application we give an asymptotic result on the spindecomposition arising in TQFT.

A quantum ergodic theorem for mapping class groups action on character variety