Entropy rate of product of independent processes

Abstract We study the multiplicative version of classical Furstenberg’s filtering problem, where instead sum $$\textbf{X}+\textbf{Y}$$ X+Y one considers product $$\textbf{X}\cdot \textbf{Y}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">X·Y ( $$\textbf{X}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">X and $$\textbf{Y}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Y are bilateral, real, finitely-valued, stationary independent processes, is taking values in $$\{0,1\}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">{0,1} ). provide formulas for $${\textbf{H}}\,(\textbf{X}\cdot \textbf{Y}\,\,|\,\,\textbf{Y})$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">H(X·Y|Y) . As a consequence, we show that if $${\textbf{H}}\,({\textbf{X}})>{\textbf{H}}\,({\textbf{Y}})=0$$ />(X)>H(Y)=0 $$\textbf{X}\amalg xmlns:mml="http://www.w3.org/1998/Math/MathML">X⨿Y , then $$\textbf{H}\,(\textbf{X}\cdot \textbf{Y})<{\textbf{H}}\,({\textbf{X}})$$ />(X·Y)<H(X) (and thus cannot be filtered out from ) whenever not bilaterally deterministic, ergodic first return to 1 can take arbitrarily long with positive probability. On other hand, almost surely visits along an infinite arithmetic progression fixed difference (with possibly some more between) find deterministic such \textbf{Y})={\textbf{H}}\,({\textbf{X}})$$ />(X·Y)=HB -free system $$(X_\eta ,S)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">(Xη,S) proximal only there always entropy drop $$h(\kappa *\nu _\eta )<h(\kappa )$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">h(κ∗νη)<h(κ) any $$\kappa $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">κ corresponding non-bilaterally process entropy. These results partly settle open problems on invariant measures systems.