Folding and Metric Entropies for Extended Shifts

In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems $({\mathrm{\Sigma }}_{m_{-},m_{+}},{\sigma }_{\phi })$ which generalizes the standard bilateral Bernoulli shifts. The space ${\mathrm{\Sigma }}_{m_{-},m_{+}}$ consists of symbolic sequences over two distinct finite alphabets, with dynamics governed by a shift map ${\sigma }_{\phi }$ incorporating a non-invertible function $\phi$ that maps one of the alphabets to the other one. These systems are, for instance, particularly useful for encoding the many-to-one baker’s transformation endomorphisms, and they can also be seen as a skew product with a unilateral Bernoulli shift on the base.