On Some Properties of Topological Entropy and Topological Pressure of Families of Dynamical Systems Continuously Depending on a Parameter

Abstract

For each everywhere dense subset \({\cal G}\) of type G_δ in a complete metric separable zero-dimensional space, we construct a family of dynamical systems continuously depending on a parameter varying in this space such that the set of points of lower semicontinuity of the topological entropy of its systems treated as a function of the parameter coincides with the set \({\cal G}\). For a family of dynamical systems continuously depending on the parameter, we prove that the set of points of lower semicontinuity and the set of points of upper semicontinuity of the topological pressure of its systems treated as a function of the parameter are sets of type G_δ and F_σδ, respectively.