Random Subsets of Self‐affine Fractals

Abstract

We find the almost sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

Dimensions of random affine code tree fractals

“…In their model the linear parts of the affine mappings are nonrandom, but the translation vectors and the number of maps are independently, randomly chosen at every step of the construction. In [13] Falconer and Miao studied random subsets of a fixed self-affine fractal. The randomness is introduced by choosing at each step of the construction a random subfamily of the original function system independently.…”

The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals

“…Recently, analogous results were obtained for random affine code tree fractals [9] and for uniformly random self-similar fractals [8]. Thereby, the natural problem arises: is it possible to compute almost sure Hausdorff dimensions for subsets of random fractals like in [7][8][9], provided the subsets are defined in terms of the limit behavior of the corresponding empirical measures (say, as in Formula (5))? One more possible direction to generalize the results of the paper is the case when the distribution ν, empirical measures δ x,n and/or the scaling factors θ(i) are defined not on the set of colors Ω, but on the whole set of infinite strings Ω N .…”

“…Falconer and Miao started the investigation of random self-affine fractals [7]. They considered a random iterated function system (RIFS) generated by a finite collection of contracting affine maps S i : R n → R n and encoded by a random tree T ∞ that is defined exactly in the same manner as the random set of infinite genetic lines X ∞ above.…”

“…Recently, the research was extended to the case of random self-affine fractals in R n (see [7,8]) and random affine code tree fractals [9]. On the other hand, Billingsley in [10,11] considered fractals in the space of infinite sequences Ω N , where Ω is a finite set, defined in terms of the limit behavior of empirical measures.…”

Geometry of spectral pairs

“…More generally, recently affine iterated function systems (aIFSs), and scale renormalization, have proved to be successful models in a variety of applications involving more difficult scaling/renormalization limits. For aIFS measures and their support, there are relatively easy formulas, see e.g., [11,14,33].…”