On factor maps that send Markov measures to Gibbs measures (Q616237)

Let \(X\) and \(Y\) be mixing shifts on finite type. Let \(\pi\) be a factor map from \(X\) to \(Y\) that is fiber-mixing, i.e., given \(x, \tilde{x} \in X\) with \(\pi(x)=\pi(\tilde{x})=y \in Y\), there is \(z\in \pi^{-1}(y) \) that is left asymptotic to \(x\) and right asymptotic to \(\tilde{x}.\) The author shows that a fiber-mixing condition is sufficient for the factor map to send all Markov measures to a Cibbs measures. The fiber-mixing condition turned out to be conjugacy-invariant and time -symmetric. It is presented an example showing that this generalization is nonempty.