Dynamical system on infinite matrices

Question

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!

Answer 1

For discrete time:

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.

For continuous time:

This type of maps appear in coagulation-fragmentation equations, as the linear part of the infinite-dimensional vector field (so as a quite simple-minded model of certain chemical reaction and modelling clusters that either separate or get together). Problems such as whether the material disseminates or whether tends to a large cluster are typical.

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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.