The following construction provides an example of an infinite network (featured on the cover of the book), which Dirichlet-to-Neumann operator satisfies the operator equation in the title of this chapter.

The matrix equation reflects the self-duality and self-symmetry of the network.

**Exercise (**).** Prove that the Dirichlet-to-Neumann operator of the network on the picture w/the natural boundary satisfies the equation.

(Hint:) Use the fact that the operator/matrix is the fixed point of the Schur complement:

where

is the circulant matrix, w/