Entropy rate of product of independent processes

Abstract

We study the multiplicative version of the classical Furstenberg's filtering problem, where instead of the sum $\mathbf{X}+\mathbf{Y}$ one considers the product $\mathbf{X}\cdot \mathbf{Y}$ ($\mathbf{X}$ and $\mathbf{Y}$ are bilateral, real, finitely-valued, stationary independent processes, $\mathbf{Y}$ is taking values in $\{0,1\}$). We provide formulas for $\mathbf{H}(\mathbf{X}\cdot\mathbf{Y}|\mathbf{Y})$. As a consequence, we show that if $\mathbf{H}(\mathbf{X})>\mathbf{H}(\mathbf{Y})=0$ and $\mathbf{X}\amalg \mathbf{Y}$, then $\mathbf{H}(\mathbf{X}\cdot \mathbf{Y})<\mathbf{H}(\mathbf{X})$ (and thus $\mathbf{X}$ cannot be filtered out from $\mathbf{X}\cdot\mathbf{Y}$) whenever $\mathbf{X}$ is not bilaterally deterministic, $\mathbf{Y}$ is ergodic and $\mathbf{Y}$ first return to $1$ can take arbitrarily long with positive probability. On the other hand, if almost surely $\mathbf{Y}$ visits $1$ along an infinite arithmetic progression of a fixed difference (with possibly some more visits in between) then we can find $\mathbf{X}$ that is not bilaterally deterministic and such that $\mathbf{H}(\mathbf{X}\cdot\mathbf{Y})=\mathbf{H}(\mathbf{X})$. As a consequence, a $\mathscr{B}$-free system $(X_\eta,S)$ is proximal if and only if there is always an entropy drop $h(\kappa\ast\nu_\eta)<h(\kappa)$ for any $\kappa$ corresponding to a non-bilaterally deterministic process of positive entropy. These results partly settle some open problems on invariant measures for $\mathscr{B}$-free systems.