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## A. Algorithm for BK Dynamics 

In order to derive a measure preserving algorithms, the first step, following Eq. (5), is to introduce a splitting of H $^{BK}$: 

<img src='content_image/366012.jpg'>

<img src='content_image/366013.jpg'>

<img src='content_image/366014.jpg'>

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<img src='content_image/366016.jpg'>

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A measure-preserving splitting of the Liouville operator then follows from Eq. (6): 

<img src='content_image/366018.jpg'>

<img src='content_image/366019.jpg'>

<img src='content_image/366020.jpg'>

<img src='content_image/366021.jpg'>

<img src='content_image/366022.jpg'>

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Upon choosing a symmetric Trotter factorization of the BL Liouville operator based on the decomposition 

a measure-preserving algorithm can be produced in full generality. 

<img src='content_image/366011.jpg'>

In practice, a choice of K$_{1}$ , K$_{2}$ , G$_{1}$ , G$_{2}$ must be made in order obtain explicit formulas. 

