A Generalization of Carateodory’s Construction for Dimensional Characteristic of Dynamic Systems

Abstract

In this paper we propose a generalization of the classical Carateodory’s construction of various characteristics of dimension type. Our approach allows us to obtain well-known dimensions (for example, the Hausdorff dimension) as well as new ones. In particular, for invariant sets of dynamical systems we define a class of dimensions which depend on the dynamics of the system. This class includes such well-known characteristics as topological pressure and topological entropy, and also a new notion: the dimension with respect to the map. It seems to me that the latter one can be used for the description of the topological and geometrical structure of the invariant set as well as the Hausdorff dimension. In multidimensional case we obtain the for mulae, connecting this dimension with the characteristics of trajectory instability of the dynamical system (such as Lyapunov exponents). These results were obtained for the two-dimensional case in [7]. Our construction deals with continuous maps of non-compact subsets of compact metric spaces. Therefore, we can also consider discontinuous maps (they are continuous on the non-compact set which does not contain preimages of the discontinuity set), for example, one-dimensional piecewise monotonic maps, Lorentz attractor (cf. [6]) and so on.