The Hausdorff dimension of some snowflake-like recursive constructions

On Hausdorff Dimension of Random Fractals

We study random recursive constructions with finite “memory” in complete metric spaces and the Hausdorff dimension of the generated random fractals. With each such construction and any positive number β we associate a linear operator V (β) in a finite dimensional space. We prove that under some conditions on the random construction the Hausdorff dimension of the fractal coincides with the value...

Random closed sets viewed as random recursions

It is known that the box dimension of any Martin-Löf random closed set of {0, 1}N is log2( 43 ). Barmpalias et al. [Journal of Logic and Computation, Vol. 17, No. 6 (2007)] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for comp...

When Thue-morse Meets Koch

In this paper, we reveal a remarkable connection between the Thue-Morse sequence and the Koch snowflake. Using turtle geometry and polygon maps, we realize the Thue-Morse sequence as the limit of polygonal curves in the plane. We then prove that a sequence of such curves converges to the Koch snowflake in the Hausdorff metric. In the final section we consider generalized Thue-Morse sequences an...

The Hausdorff Dimension and Exact Hausdorff Measure of Random Recursive Sets with Overlapping

Random Recursive Triangulations of the Disk via Fragmentation Theory

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact su...