Abstract
The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their digits. All our results are of the following form:a typical (in the sense of Baire) point xin K has the following property: the average frequencies of digits of x have maximal oscillation.We consider general types of average frequencies, namely, average frequencies associated with general averaging systems. These averages include, for example, all higher order Hölder and Cesaro averages, and Riesz averages. Surprising, for all averaging systems (regardless of how powerful they are) we prove that a typical (in the sense of Baire) point xin K has the following property: the average frequencies of digits of x have maximal oscillation. This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible. Several applications are given, e.g. to continued fraction digits and Lüroth expansion digits.
Highlights
Self-similar sets are arguably the most important examples of fractal sets
Lower bounds for the Hausdorff and packing dimensions of self-similar sets are invariably obtained by using the ergodic theorem to examine the limiting distribution of almost all frequencies of digits; these results are translated into lower bounds for the fractal dimension using the “distribution of mass” principle
The detailed study of almost sure frequencies of digits of different expansions is a fundamental area in metric number theory with deep connections to, for example, Diophantine approximations, see [14] and [5]
Summary
Self-similar sets are arguably the most important examples of fractal sets. This is due to their simple description and the fact that they exhibit many of the properties one expects from fractals. Short studies of the distribution digits of typical points belonging to self-similar sets associated with IFS’s and IIFS’s are present in [2,17,18]. From this that a typical (in the sense of Baire) point x has the following property: all higher order Hölder averages and all higher order Cesaro averages of digits of x have maximal oscillation This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible.
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