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## Appendix A: Operator formula 

We wish to determine the action of the propagator associated with the Liouville operator Eq. (39). This is equivalent to solving the evolution equation (recall i ̸ = k ) 

from t = 0 to t= τ . Integrating, we have 

giving 

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## Appendix B: Invariant Measure of the BK phase space flows 

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The phase space compressibility of the phase space BK thermostat is 

<img src='content_image/336993.jpg'>

Upon introducing the function 

one can easily find that 

so that the invariant measure in phase space reads 

as desired. 

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