Geometry of measures in random systems with complete connections (Q2120872)

One can cite authors' description of this research: ``In this paper, we study relations between countable conformal iterated function systems (IFS) with arbitrary overlaps, Smale's endomorphisms, and random systems with complete connections, from the point of view of their geometric and ergodic properties. We provide a common framework for studying measures with certain invariance properties and their dimensions in these systems. We prove that stationary measures for countable conformal IFS with overlaps and place-dependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps \(\mathcal{S}\), and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system \(\mathcal{S}\).'' In the survey of this paper, such notions as conformal iterated function systems, finite iterated function systems with place-dependent probabilities (weights), and random systems with complete connections, as well as finite conformal IFS with overlaps, countable conformal IFS with overlaps, and finite iterated function systems with place-dependent probabilities, etc., are noted. One can remark that the present research deals with the following notions: countable IFS \(\mathcal{S}\) with place-dependent probabilities and arbitrary overlaps, an arbitrary countable IFS with overlaps \(\mathcal{S}\) which satisfies a condition of pointwise non-accumulation, and random systemswith complete connections, as well as Smale's skew product endomorphism, etc. Also, the notion of finite IFS with place-dependent probabilities is extended to countable iterated function systems with overlaps and place-dependent probabilities.