Targets, local weak $σ$-Gibbs measures and a generalized Bowen dimension formula

Abstract: For a dynamical system, we study the set of points $\cal W$ whose orbit approximates any chosen point at certain specified rates. Our basic setting is that of left shift acting on topological Markov chains endowed with a local weak Gibbs measure. Our rates of recurrence are so fast that the corresponding set $\cal W$ has measure zero, but we obtain a generalized Bowen formula for Carath\'eodory dimension. For the case of Markov transformations with countable partition and big image (BI) property a Bowen-type formula is obtained for the Hausdorff dimension of those exceptional sets. In particular, we apply our general results to Gauss and Luroth maps, a not Bernoulli modification of Gauss map and some inner functions. Since we only require the existence of weak Gibbs measures we can deal with non H\"older potentials and we can also consider intermittent systems as the Manneville-Pomeau map.