Let (X, d, f ) be a dynamical system, where (X, d) is a compact metric space and f : X → X is a continuous map. Using the concepts of g-almost product property and uniform separation property introduced by Pfister and Sullivan in [22], 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, β-shifts, repellers and uniformly hyperbolic diffeomorphisms.

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence under a mild assumption we prove exponential large deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

We define the notion of sequential Gibbs measures, inspired by on the classical notion of Gibbs measures and recent examples from the study of non-uniform hyperbolic dynamics. Extending previous results of and Ugalde-Chazottes [2], we show that the images of one block factor maps of a sequential Gibbs measure are also a sequential Gibbs measure, with the same sequence of Gibbs times. We obtain some estimates on the regularity of the potential of the image measure at almost every point.

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