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- W4287814532 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 $mathbf{H}(mathbf{X}cdotmathbf{Y}|mathbf{Y})$. 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 $mathbf{X}cdotmathbf{Y}$) 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 $mathbf{H}(mathbf{X}cdotmathbf{Y})=mathbf{H}(mathbf{X})$. As a consequence, a $mathscr{B}$-free system $(X_eta,S)$ is proximal if and only if there is always an entropy drop $h(kappaastnu_eta)<h(kappa)$ 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." @default.
- W4287814532 created "2022-07-26" @default.
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- W4287814532 date "2020-04-16" @default.
- W4287814532 modified "2023-09-24" @default.
- W4287814532 title "Entropy rate of product of independent processes" @default.
- W4287814532 doi "https://doi.org/10.48550/arxiv.2004.07648" @default.
- W4287814532 hasPublicationYear "2020" @default.
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