Measure of intermediate local/pointwise dimension is not ergodic for $T_\beta(x)=\beta \cdot x \mod 1$ if $\beta \not \in \mathbb{N}$

Question

Let $\mu$ be an ergodic measure for the map $T_a(x)=a \cdot x \mod 1$ for $a \in \mathbb{N}$ with $a \geq 2$ and invariant for $T_{a^{\frac{1}{n}}}$ for some $n \in \mathbb{N}$ (and thus also ergodic). By assumption $\mu$ has intermediate dimension, which means that $\lim_{r \to 0}\frac{\log \mu(B_r(x))}{\log r}$ exists for $\mu$-almost every $x$ and the value is the same $\mu$-almost everywhere with the almost sure value is in $(0,1)$. Is it true that this implies that $a^{\frac{1}{n}} \in \mathbb{N}$? I don't see why a measure of intermediate dimension can't be ergodic for $T_\beta$ with $\beta \not \in \mathbb{N}$.