Local dimensions of self-similar measures satisfying the finite neighbour condition*

Abstract

We study sets of local dimensions for self-similar measures in R satisfying the finite neighbour condition, which is formally stronger than the weak separation condition (WSC) but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. We also explain how our results allow computations of the sets of local dimensions in many explicit cases. This contextualises and generalises a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

“…Lau and Ngai [43] defined the WSC in slightly different form to analyse the multifractal structure of one-dimensional overlapping self-similar measures, following Lalley [42]. One-dimensional WSC attractors and measures have been further investigated by Feng [27,28], Hare and Hare with coauthors [35,36,37] and others [21,40]. There are few studies of two-dimensional WSC examples [12,19,64], and apparently no studies for greater dimension.…”

“…These links form the neighbor graph. See [8,10,11,12,37,45,46,58,63] for details and examples. Overlap neighbors are those which admit an outgoing path to the initial vertex id.…”

“…Only in 1962, this was clarified by Garsia [32] and led to a lot of research in onedimensional self-similar measures, cf. [3,27,28,31,35,37,42,43]. In 1982, Rauzy found a connection between substitutions with a matrix with Pisot eigenvalue, dynamical systems and self-similar tilings [57] which triggered much work on fractal tilings [1,4,51,56] and the still unsolved Pisot substitution conjecture [2,13,16,50].…”

“…The two neighbors with one-point intersection are neglected. For one-dimensional attractors, this makes sense and is taken as FT definition in recent papers by Hare and coauthors [36,37].…”

“…The formula with the spectral radius of the neighborhood incidence matrix M greatly improved Zerner's approximation method. In dimension one, M is easily determined when the set V is an interval [37]. In dimension greater one, however, the graph of neighborhoods of an open set V is very difficult to establish, unless A is as simple as Figure 4.…”