Chaotic sets and Hausdorff dimension for L\"uroth expansions

We provide new similarities between regular continued fractions and L"uroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the L"uroth transformation of results by W. Liu, B. Li and W. Liu, S. Wang on the distal, asymptotic and Li-Yorke pairs for the Gauss map.