具有独立孤子分量的BBS不变测度
BBS invariant measures with independent soliton components
摘要：
盒球系统（BBS）是由Takahashi和Satsuma \cite{TS}引入的$\{0,1\}^\Z$中的一维元胞自动机，他们还确定了称为\emph{solitons}的保守序列。整数称为盒子，球配置表示球所占据的盒子。对于每个整数$k\ge1$，$k$ -孤子由被球占据的$k$盒和$k$空盒（不一定是连续的）组成。Ferrari、Nguyen、Rolla和Wang \cite{FNRW}将配置的$k$ -槽定义为可以插入$k$ -孤子的位置。用整数标记$k$ -槽，它们将配置的$k$ -分量定义为$\Z_{\ge0}$的元素的阵列$\{ζ_k(j)\}_{j\in \mathbb Z}$，给出附加到$k$ -槽$j\in \mathbb Z$的$k$ -孤子的数目$ζ_k(j)$。他们还表明，如果棕榈变换的平移不变分布$μ$有独立的孤子组件，那么$μ$是不变的自动机。我们证明了对于每个$λ\in[0,1/2)$的Palm变换的乘积Bernoulli测量参数$λ$有独立的孤子组件和它的$k$ -组件是几何随机变量的乘积测量参数$1-q_k(λ)$，$λ$的显式函数。该结构用于描述Palm变换下具有独立分量的一大族不变测度，包括Markov度量。
英文摘要：
The Box-Ball System (BBS) is a one-dimensional cellular automaton in $\{0,1\}^\Z$ introduced by Takahashi and Satsuma \cite{TS}, who also identified conserved sequences called \emph{solitons}. Integers are called boxes and a ball configuration indicates the boxes occupied by balls. For each integer $k\ge1$, a $k$-soliton consists of $k$ boxes occupied by balls and $k$ empty boxes (not necessarily consecutive). Ferrari, Nguyen, Rolla and Wang \cite{FNRW} define the $k$-slots of a configuration as the places where $k$-solitons can be inserted. Labeling the $k$-slots with integer numbers, they define the $k$-component of a configuration as the array $\{ζ_k(j)\}_{j\in \mathbb Z}$ of elements of $\Z_{\ge0}$ giving the number $ζ_k(j)$ of $k$-solitons appended to $k$-slot $j\in \mathbb Z$. They also show that if the Palm transform of a translation invariant distribution $μ$ has independent soliton components, then $μ$ is invariant for the automaton. We show that for each $λ\in[0,1/2)$ the Palm transform of a product Bernoulli measure with parameter $λ$ has independent soliton components and that its $k$-component is a product measure of geometric random variables with parameter $1-q_k(λ)$, an explicit function of $λ$. The construction is used to describe a large family of invariant measures with independent components under the Palm transformation, including Markov measures.
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