located at an odd position, and also a stochastic update on the last site $L$ (describing an interaction with reservoir at the right boundary). ${{\mathbb U}}^{e}$ similarly realizes a stochastic update on every pair of consecutive sites with the first site located at an even position, and also a stochastic update on the first site $1$ (describing an interaction with reservoir at the left boundary). Note that the operators ${{\mathbb U}}^{o}$ and ${{\mathbb U}}^{e}$ do not commute and can be interpreted as describing a two-step Floquet dynamics. A pictorial representation of the discrete time dynamics is given in figure \[fig:open\]. This discrete time update is very similar to the *sublattice-parallel update* [@RajewskySSS]: the difference is that in the present model the size of the lattice $L$ has to be *odd* instead of *even* in the sublattice-parallel update with open boundaries.

iin [0,2,...,8]{} [(i,0) – (i,0.5) ; (i,0.5) – (i,1) ; (i,2) – (i,3) ; (i,4) – (i,5) ; (i,6) – (i,6.5) ; (i,6.5) – (i,7) ;]{} iin [1,2,5,6]{} [(-0.5,i) – (0.5,i) ; (1.5,i) – (4.5,i) ; (5.5,i) – (8.5,i) ;]{} iin [3,4]{} [(-0.5,i) – (2.5,i) ; (3.5,i) – (6.5,i) ; (7.5,i) – (8.5,i) ;]{} iin [-0.5,0.5,1.5,4.5,5.5,8.5]{} [(i,1) – (i,2) ; 