I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out:
Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n \}$ and $\mu$ a measure on $X$ that is $T \colon X \ni x \to (px\mod 1)$ invariant. We define the measure $$\omega_n = \sum_{\alpha \in D_n} \mu * \delta_\alpha = \sum_{\alpha \in D_n} \mu(\cdot + \alpha ) $$ Notice that $\mu = \omega_0 \ll \cdots \ll \omega_n \ll \omega_{n+1}$ then the Radon Nikodym derivative $\phi_n = \frac{d\mu}{d\omega_n}$ exists and the paper said that is easy to see that $$\phi_n = \prod_{k=0}^{n-1}\phi_1 \circ T^k$$ but I could not do it, although I reduced it to the following:
If we proove this $$\frac{d\omega_{n}}{d\omega_{n+1}} = \frac{d\omega_{n-1}}{d\omega_n} \circ T$$ we can use the chain rule for Radon Nikodym derivatives to conclude.
Any help will be appreciated.