Does full shift have the local product structure?

Question

We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is of the form $$ \mu_{|[I]}=\psi(\mu^{+} \times \mu^{-}),$$ where $\psi$ is continuous and positive, and $\mu^{+}$ and $\mu^{-}$ are the projections of $\mu_{|[I]}$ to the spaces of one-sided sequences indexed by positive and negative integers, respectively.

Let $T:\Sigma \to \Sigma$ be a full shift. Are all $T-$ invariant measures have the local product structure?if not, could you please give an example ?

Update: Let $f:M \to M$ be a $C^{1+\alpha}$ diffeomorphism. Is it true that all hyperbolic measures have local product structure?

Answer 1

No. Take $\mu$ to be a measure supported on a Sturmian shift corresponding to some irrational rotation $R_\alpha$. If $\mathcal F^+$ denotes the $\sigma$-algebra generated by the coordinates in $\mathbb Z_{\ge 0}$ and $\mathcal F^-$ denotes the $\sigma$-algebra generated by coordinates in $\mathbb Z_{<0}$, then conditional on $\mathcal F^+$, $\mu$ is supported on a single point (there is an almost everywhere one-one correspondence between negative and positive tails). This means $\mu$ is very far from a product of $\mu^-$ and $\mu^+$ (in which for each positive tail, all negative tails occur with conditional probability $\psi$).