Discrete Triebel-Lizorkin spaces and expansive matrices

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$ and $B$, it is shown that ${\dot{\mathbf{f}}}_{p,q}^{\alpha }(A)={\dot{\mathbf{f}}}_{p,q}^{\alpha }(B)$ for all $\alpha \in \mathbb{R}$ and $p,q\in (0,\mathrm{\infty }]$ if and only if the set $\{A^j{B}^{-j}:j\in \mathbb{Z}\}$ is finite, or in the trivial case when $p=q$ and $\mathrm{\mid }\det (A){\mathrm{\mid }}^{\alpha +1\mathrm{/}2-1\mathrm{/}p}=\mathrm{\mid }\det (B){\mathrm{\mid }}^{\alpha +1\mathrm{/}2-1\mathrm{/}p}$. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.