Variational principles on subsets of non-autonomous dynamical systems: topological pressure and topological entropy

This paper discusses the variational principles on subsets for topological pressure and topological entropy of non-autonomous dynamical systems. We define the Pesin-Pitskel topological pressure (weighted topological pressure) and the Bowen topological entropy (weighted Bowen topological entropy) for any subset. Also, we define the measure-theoretic pressure and the measure-theoretic lower entropy for all Borel probability measures. Then, we prove variational principles for topological pressure (topological entropy) which links the Pesin-Pitskel topological pressure (weighted topological pressure) on an arbitrary nonempty compact subset to the measure-theoretic pressure of Borel probability measures for non-autonomous dynamical systems (which links the Bowen topological entropy (weighted Bowen topological entropy) on an arbitrary nonempty compact subset to the measure-theoretic lower entropy of Borel probability measures for non-autonomous dynamical systems). Moreover, we show that the Pesin-Pitskel topological pressure (weighted topological pressure) and the Bowen topological entropy (weighted Bowen topological entropy) can be determined by the measure-theoretic pressure and the measure-theoretic lower entropy of Borel probability measures, respectively. These results extend Feng and Huang's results (Variational principles for topological entropies of subsets, J. Funct. Anal. (2012)), Ma and Wen's results (A Billingsley type theorem for Bowen entropy, Comptes Rendus Mathematique (2008)), and Tang et al. results (Variational principle for topological pressures on subsets, J. Math. Anal. Appl. (2015)) for classical dynamical systems to pressures and entropies of non-autonomous dynamical systems.