The Asymptotically Additive Topological Pressure: Variational Principle For Non Compact and Intersection of Irregular Sets

Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space and $f:X\to X$ is a continuous map. Using the concepts of $\mathit{\text{g-almost}}\mathit{\text{product}}\mathit{\text{property}}$ and $\mathit{\text{uniform}}\mathit{\text{separation}}\mathit{\text{property}}$ introduced by Pfister and Sullivan in \cite{Pfister2007}, we give a variational principle for certain non-compact with relation the asymptotically additive topological pressure. We also study the set of points that are irregular for an collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure. These results are suitable for systems such as mixing shifts of finite type, $\beta$-shifts, repellers and uniformly hyperbolic diffeomorphisms.