This question is related to my question:

entropy preserving finitary factor maps of Bernoulli schemes.

Hopefully, this one is a bit easier.

Let $X=\{0,1\}^\mathbb{Z}$ with measure $\mu=(p,1-p)^{\mathbb{Z}}$, where $p \not = 1/2$.

For $x \in X$, define $x^*$ so that $x^*_i=(x_i+1)$mod$2$.

A factor map $\psi$ is finitary if for almost every $x \in X$ there exists integers $m \leq n$ such that the zero coordinates of $\psi(x)$ and $\psi(x')$ agree for almost all $x' \in X$ with $x[m,n]=x'[m,n]$. In other words, a factor map is finitary if it is continuous after removing a subset of measure zero.

Let $\psi:X \to Y$, such that for almost all $x \in X$, $\psi(x)=\psi(x^*)$.

Can we find an example of such a $\psi$ and $Y$ where $h(Y)=h(X)$ ($\psi$ does not decrease entropy), and $Y$ is Markov (or even variable length Markov)?

Thank you.