Triebel-Lizorkin spaces and expansive matrices

We survey recent classification theorems for expansive matrices that generate the same anisotropic homogeneous Triebel-Lizorkin function space or sequence space. The function spaces are classified precisely by those matrices for which their associated homogeneous quasi-norms on Euclidean space are equivalent, whereas the sequence spaces are classified by a strictly stronger condition. We unravel this discrepancy between function spaces and sequence spaces by showing that two sequence spaces are retracts of each other whenever the corresponding function spaces are the same.