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SUMMARY:Joanna Kułaga-Przymus (UMK Toruń)
DTSTART:20201009T081500Z
DTEND:20201009T094500Z
DTSTAMP:20260423T021930Z
UID:DSSUJ/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/DSSUJ/10/">E
 ntropy rate of product of independent processes</a>\nby Joanna Kułaga-Prz
 ymus (UMK Toruń) as part of Dynamical systems seminar at the Jagiellonian
  University\n\nLecture held in 1016.\n\nAbstract\nThe entropy of the produ
 ct of stationary processes is related to\nFurstenberg’s filtering proble
 m. In its classical version one deals\nwith the sum $\\bm{X}+\\bm{Y}$\, wh
 ere $\\bm{X}$ corresponds to the signal\nand $\\bm{Y}$ to the noise. In hi
 s seminal paper from 1967\, Furstenberg\nshowed that under the natural ass
 umption of the disjointness of\nunderlying dynamical systems\, the informa
 tion about $\\bm{X}$ can be\nretrieved from $\\bm{X}+\\bm{Y}$. Instead of 
 the sum\, we study the\nproduct $\\bm{X}\\cdot\\bm{Y}$. We give a formula 
 for the entropy rate of\n$\\bm{X}\\cdot\\bm{Y}$ (relative to that of $\\bm
 {Y}$\, for $\\bm{X}$ and\n$\\bm{Y}$ being independent). As a consequence\,
  $\\bm{X}$ cannot be\nrecovered from $\\bm{X}\\cdot\\bm{Y}$ for a wide cla
 ss of positive\nentropy processes\, including exchangeable processes\, Mar
 kov chains and\nweakly Bernoulli processes. Moreover\, we answer some open
  problems on\nthe dynamics of $\\mathscr{B}$-free systems (including the s
 quare-free\nsystem given by the square of the Moebius function). The talk 
 is based\non joint work with Michał Lemańczyk\, see\nhttps://arxiv.org/p
 df/2004.07648.pdf\n
LOCATION:https://researchseminars.org/talk/DSSUJ/10/
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