On some properties of the local entropy of dynamical systems unwinded as a function of a point in the phase space

The article studies the nature of the dependence on the local entropy point of a non-autonomous dynamical system. It is proved that the local entropy is a function of the second Baire class on the phase space, and its set of lower semicontinuity points forms an everywhere dense set of type $G_\delta$. An autonomous dynamical system is constructed such that the set of above semicontinuity points of the local entropy of this system is empty.