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International audienceWe show that the Teichmüller space of a surface without boundary and with punctures, equipped with the Thurston metricis the limit in an appropriate sense of  Teichmüller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances of mapping classes for their actions on Teichmüller spaces equipped with the Thurston metric

Thurston's metric on Teichmüller space and the translation distances of mapping classes