Factor Complexity of the Most Significant Digits of~$a^{n^d}$

We investigate unipotent dynamics on a torus and apply these techniques to the following problem. Let $d$ be a positive integer, and let $a > 0$ be a real number. For an integer $b \geqslant 5$, such that $a$ and $b$ are multiplicatively independent, consider the sequence $(\mathbf{w}_n)$, where $\mathbf{w}_n$ is the most significant digit of $a^{n^d}$ when expressed in base $b$. We prove that the complexity function of the sequence $(\mathbf{w}_n)$ is, up to finitely many exceptions, a polynomial function.