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تسجيل مستخدم جديدBBS invariant measures with independent soliton components
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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 $kge1$, 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 ${zeta_k(j)}_{jin mathbb Z}$ of elements of $Z_{ge0}$ giving the number $zeta_k(j)$ of $k$-solitons appended to $k$-slot $jin mathbb Z$. They also show that if the Palm transform of a translation invariant distribution $mu$ has independent soliton components, then $mu$ is invariant for the automaton. We show that for each $lambdain[0,1/2)$ the Palm transform of a product Bernoulli measure with parameter $lambda$ has independent soliton components and that its $k$-component is a product measure of geometric random variables with parameter $1-q_k(lambda)$, an explicit function of $lambda$. 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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