Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.

Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where $a,b$ are given sequences in $\mathbb{Z}_{2}^{\mathbb{N}}$. Consider the set $M(\sigma \times \sigma)$ of $\sigma \times \sigma$ invariant probability measures on $\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}}$. Here $\sigma$ is the full-shift.

**Question: Given $\eta, \mu$ $\sigma$-invariant measures, how big is the set
$$A_{\eta,\mu}=\{ \rho \in M(\sigma \times \sigma): \rho \circ S^{-1} = \eta*\mu\}?$$**

**Obs:** I know that $A_{\eta,\mu}$ is non-empty, because $(\eta \times \mu) \in A_{\eta,\mu}$. The $*$ is the convolution of two measures and can be defined on this way:

$$\eta*\mu:=(\eta \times \mu) \circ S^{-1}.$$

I would like to understand the set $A_{\eta,\mu}$ deeply and, for instance, to see some examples of elements of it.