An easy question that I have never been able to answer. Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A_{x,y}$ and $A$ the $3\times 3$ matrix $A=(0,1,2,0,1,2,1,2,0).$ For example $(0,1,2,0,0,0,1,2,0,\ldots)\mapsto (1,2,1,0,0,1,2,1,\ldots),$ It is like a shift if the coordinate before is $0$ or $1$ and $x+1$ mod 3 if not. My question is, are there any invariant probability measures full supported other that the Haar measure? I have never seen an idea to solve this problem, so any reference is welcome.
An easy question that I have never been able to answer. Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A_{x,y}$ and $A$ the $3\times 3$ matrix $A=(0,1,2,0,1,2,1,2,0).$ For example $(0,1,2,0,0,0,1,2,0,\ldots)\mapsto (1,2,1,0,0,1,2,1,\ldots),$ It is like a shift if the coordinate before is $0$ or $1$ and $x+1$ mod 3 if not. My question is, are there any invariant probability measures other that the Haar measure? I have never seen an idea to solve this problem, so any reference is welcome.