The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets

In this paper, we consider the planar self-affine measures ${\mu }_{M,D}$ generated by an expanding matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $ D=\left{ {\left( {\begin{array}{{20}{c}} 0\ 0 \end{array}} \right),\left( {\begin{array}{{20}{c}} \alpha_1\ \alpha_2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} \beta_1\ \beta_2 \end{array}} \right)} \right} $with$\alpha_1\beta_2-\alpha_2\beta_1\neq0$\mathrm{.}Weshowthatif$\det(M)\notin 3$\mathbb{Z}$$, then the mutually orthogonal exponential functions in $L^2({\mu }_{M,D})$ is finite, and the exact maximal cardinality is given.