论文标题
独立过程产品的熵率
Entropy rate of product of independent processes
论文作者
论文摘要
我们研究经典Furstenberg的过滤问题的乘法版本，其中而不是总和$ \ Mathbf {x}+\ \ \ \ \ \ Mathbf {y} $一个人认为产品$ \ Mathbf {x} \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ mathbf {y}有限价值的固定独立流程，$ \ mathbf {y} $以$ \ {0,1 \} $为中。我们为$ \ mathbf {h}提供公式（\ Mathbf {x} \ cdot \ mathbf {y} | \ mathbf {y}）$。结果，我们表明，如果$ \ MATHBF {h}（\ MathBf {X}）> \ MathBf {h}（\ MathBf {y}）= 0 $和$ \ Mathbf {x}} \ amalg \ amalg \ amalg \ amalg \ Mathbf \ Mathbf {y} $，然后\ Mathbf {y}）<\ MathBf {h}（\ MathBf {X}）$（因此，$ \ Mathbf {X} $无法从$ \ MathBf {x} \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ mathbf {y} $ nesivical n n n n n n n n n nes nestion nesigation中$ \ mathbf {y} $是ergodic，$ \ mathbf {y} $首先返回到$ 1 $，可能会以正概率为单位。另一方面，如果几乎肯定$ \ mathbf {y} $访问$ 1 $沿着无限的算术差异（之间可能有更多访问），那么我们可以找到$ \ mathbf {x} $，而不是双边确定性的和如此确定性的， $ \ mathbf {h}（\ mathbf {x} \ cdot \ mathbf {y}）= \ mathbf {h}（\ Mathbf {x}）$。结果，$ \ mathscr {b} $ - 自由系统$（x_η，s）$在且仅当总是有熵下降$ h（κ\astν_η）<h（κh（κh（κ）$的任何$κ$的任何$κ$，对与非单侧确定性的正熵过程相对应的任何$κ$。这些结果部分解决了$ \ Mathscr {B} $ - 免费系统的不变措施的一些开放问题。
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 $(X_η,S)$ is proximal if and only if there is always an entropy drop $h(κ\astν_η)<h(κ)$ for any $κ$ 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.
