# On a certain set of probability measures on a shift

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.

• Have you tried to look what happens for some examples? What happens if one of the measures have zero entropy? What happens if the measures are disjoint? (this does not happen if both of them have positive entropy). It is for example very easy to conclude some lower bound over the dimension of $\rho$ by simple entropy estimate. Maybe you can get something from the recent work of Mike Hochman. – Asaf Nov 16 '15 at 19:39
• Do you really mean $M(\sigma\times\sigma)$? or do you mean $M(\sigma)\times M(\sigma)$? – Anthony Quas Nov 16 '15 at 23:07
• Then I think the set is huge. Given any probability measure $\nu\in M(\sigma)$, define a map $\Phi$: $X\times X$, $(x,y)\mapsto (x,S(x,y))$. Now $\Phi^*(\nu,\eta*\mu)\in A_{\eta,\mu}$ for each $\nu$. – Anthony Quas Nov 16 '15 at 23:14
• If you meant the subset of $M(\sigma)\times M(\sigma)$, then I think if you take $\mu$ to be a Sturmian shift and $\eta$ to be the $\delta$-measure supported on the 0 fixed point, I think $A_{\eta,\mu}$ is extremely small. – Anthony Quas Nov 17 '15 at 2:32
• Mike proved an inverse theorem for entropy in his Annals paper - "On self-similar sets with overlaps and inverse theorems for entropy", showing what happens if the entropy of the convolution does not grow. As you're interested in given convolution, if the entropy of the convolution isn't large compered with the original entropy of $\mu$, it will limit, in a severe manner, your set (well not exactly, it will limit the joinings of $\mu$ and $\nu$). But it can be a place to start, as I've hinted in the first comment. – Asaf Nov 17 '15 at 21:28