Poincaré recurrences time and local dimension of chaotic attractors

The concept of a local fractal dimension has been introduced in the framework of the average Poincaré recurrence time numerical analysis in an $\varepsilon$-vicinity of a certain point. Lozi and Hénon maps have been considered. It has been shown that in case of Lozi map the local dimension weakly depends on the point on the attractor and its value is close to the fractal dimension of the attractor. In case of a quasi attractor observed in both Hénon and Feugenbaum systems the local dimension significantly depends on both the diameter and the location of the $\varepsilon$-vicinity. The reason of this strong dependency is high non-homogenity of a quasi-attractor which is typical for non-hyperbolic chaotic attractors.