On a kind of self-similar sets with complete overlaps

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation prop...

Hausdorff Dimension of Self-similar Sets with Overlaps

Correlation dimension for self-similar Cantor sets with overlaps

We consider self-similar Cantor sets Λ ⊂ R which are either homogeneous and Λ− Λ is an interval, or not homogeneous but having thickness greater than one. We have a natural labeling of the points of Λ which comes from its construction. In case of overlaps among the cylinders of Λ, there are some “bad” pairs (τ, ω) of labels such that τ and ω label the same point of Λ. We express how much the co...

Self-similar Sets with an Open Set Condition and Great Variety of Overlaps

For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in [0, 0.2] as well as a finite set of any cardinality 2m. The main point is that the open set condition is fulfilled for all these examples.

A kind of fuzzy upper topology on L-preordered sets

Considering a commutative unital quantale L as the truth value table and using the tool of L-generalized convergence structures of stratified L-filters, this paper introduces a kind of fuzzy upper topology, called fuzzy S-upper topology, on L-preordered sets. It is shown that every fuzzy join-preserving L-subset is open in this topology. When L is a complete Heyting algebra, for every completel...