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Assuming the existence of certain large cardinal numbers, we prove that for every projective filter F over the set of natural numbers, F‐bases in Banach spaces have continuous coordinate functionals. In particular, this applies to the filter of statistical convergence, thereby we solve a problem by Kadets (at least under the presence of certain large cardinals). In this setting, we recover also a result of Kochanek (Studia Math., 2012) who proved continuity of coordinate functionals for countably generated filters. [ABSTRACT FROM AUTHOR]

We study the concept of a code (or shift) space for a generalized iterated function system (GIFS in short). We prove that relations between GIFSs and their code spaces are analogous to the case of classical IFSs. As an application, we consider the problem of connectedness of attractors of GIFSs. Many of our results are strengthening the ones proved recently by Mihail, Miculescu and Secelean, but some are completely new. [ABSTRACT FROM AUTHOR]

We follow the idea of generalising the notion of classical iterated function systems, as presented by Mihail and Miculescu. We give their deliberations a more general setting and, using this general approach, study the generic aspect of the problem of existence of an attractor of a function system. [ABSTRACT FROM AUTHOR]