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 textbf{X}+textbf{Y} one considers the product textbf{X}cdot textbf{Y} (textbf{X} and textbf{Y} are bilateral, real, finitely-valued, stationary independent processes, textbf{Y} is taking values in {0,1}). We provide formulas for {textbf{H}},(textbf{X}cdot textbf{Y},,|,,textbf{Y}). As a consequence, we show that if {textbf{H}},({textbf{X}})>{textbf{H}},({textbf{Y}})=0 and textbf{X}amalg textbf{Y}, then textbf{H},(textbf{X}cdot textbf{Y})<{textbf{H}},({textbf{X}}) (and thus textbf{X} cannot be filtered out from textbf{X}cdot textbf{Y}) whenever textbf{X} is not bilaterally deterministic, textbf{Y} is ergodic and textbf{Y} first return to 1 can take arbitrarily long with positive probability. On the other hand, if almost surely textbf{Y} visits 1 along an infinite arithmetic progression of a fixed difference (with possibly some more visits in between) then we can find textbf{X} that is not bilaterally deterministic and such that {textbf{H}},(textbf{X}cdot textbf{Y})={textbf{H}},({textbf{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 *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.