# Dynamical system on infinite matrices

I am interested in a pair of dynamical systems, and I'm wondering whether these have a connection to something else in mathematics.

Let $$G$$ be an additive abelian group, and let $$\mathcal{M}(G)$$ denote the set of infinite matrices with entries from $$G$$; a generic element of $$\mathcal{M}(G)$$ is $$A = (a_{i,j})_{i,j = 1}^\infty$$. We can define a metric on $$\mathcal{M}(G)$$ as follows.

For each matrix $$A$$, let $$A[k] = (a_{i,j})_{i,j=1}^k$$ be the upper-left $$k\times k$$ block of $$A$$. Given any two matrices $$A$$ and $$B$$, let $$n$$ be the largest nonnegative integer such that $$A[n] = B[n]$$; we set $$d(A,B)= 1/2^n$$. If no such $$n$$ exists, set $$d(A,B) = 0$$. With this metric, $$\mathcal{M}(G)$$ is a complete metric space.

There are two maps that act on this space that I'm interested in; both send $$\mathcal{M}(G)$$ to $$\mathcal{M}(G)$$. The first I'll denote with $$\sigma$$, defined by $$\sigma(A)_{i,j} = a_{i,j} + a_{i+1,j} + a_{i,j+1} + a_{i+1,j+1}.$$ The second is $$\tau$$, defined by $$\tau(A)_{i,j} = a_{i,j} + a_{i-1,j} + a_{i,j-1} + a_{i-1,j-1},$$ where $$a_{i,j} = 0$$ if $$i$$ or $$j$$ is $$0$$.

Both maps are Lipschitz continuous, and in fact $$\tau$$ is an isometry: $$d(\tau(A),\tau(B)) = d(A,B)$$. Moreover, $$\tau$$ is invertible ($$\sigma$$ is not).

My main question is whether either of these systems is related to or reminiscent of another dynamical system or some other structure from mathematics. Thank you for your help!

Your maps are close to the averaging procedure of the Laplacian. More precisely, when we discretize the Laplacian the value at a given point is obtained taking the average on $$4$$ close points. So the iteration of the maps $$\sigma$$ and $$\tau$$ should approximate a harmonic function when the size of the discretization tends to zero. Something more precise really depends on what you want.
It is better for such models to consider $$G^{\mathbb Z\times\mathbb Z}$$ instead of $$G^{\mathbb N\times\mathbb N}$$. Also, it is often convenient to introduce a scalar product.