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 $b⩾5$, 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.