The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition

Abstract

Abstract. Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.1. Introduction. Mauldin and Williams [6] computed the Hausdorff dimension of the attractor for a Graph Directed Iterated Function System (GIFS) of similarities in R d assuming that the cylinders are, in some sense, well separated. If we drop the separation condition in the Mauldin-Williams theorem, we can expect only "almost all" type results. In this paper we prove such a result on the line assuming neither any separation conditions nor any restrictions on the contraction ratios. The only thing we have to assume is that the matrix of the directed graph is irreducible. Our research was motivated by an open problem in the field of fractal image compression. This problem is about the box-counting dimension of a certain graph directed non-conformal attractor in space. Using techniques similar to the ones in [4], we can reduce this problem to the computation of the attractor of some GIFS on the line with overlapping cylinders.

Resonance between Cantor sets

“…In general, those results are sharp in the sense that the set of exceptions may have positive, or even full, dimension. But when the constructions are done in a dynamical or geometrically regular way, there is often some countable set of parameters, usually arising from an algebraic condition, where the result fails, and it is natural to conjecture that these are all parameters for which the result fails, see for example [6], [8], [20], [18], [7]. However, there are very few cases where the set of exceptions has been explicitly determined.…”

A lower bound for the symbolic multifractal spectrum of a self‐similar multifractal with arbitrary overlaps

“…Indeed, during the past 10 years there has been a great interest in obtaining bounds for the Hausdorff dimension of self-similar sets not satisfying the OSC and a large number of results have been obtained. In particular, we would like to mention the work of Solomyak, Simon, and Peres on the transversality condition which can be applied to investigate the Hausdorff dimension of certain classes of self-similar sets not satisfying the OSC [10], [21], [23], [24].…”

“…As mentioned earlier, during the past 10 years there has been a great interest in obtaining bounds for the Hausdorff dimension of self-similar sets not satisfying the OSC and a large number of results have been obtained. In particular, the work of Solomyak, Simon and Peres and collaborators [9], [10], [20], [21], [23], [24] has been applied to investigate the Hausdorff dimension of the (0, 1, 3)-set Γ γ of γ-expansions with deleted digits. Indeed, in general, inequality (1.15) can be improved significantly.…”

Dimension of the repeller for a piecewise expanding affine map

“…This IFS contains only affine transformations with triangular matrices. The dimension theory of the interpolation functions was studied in several papers, see for example Bedford [14], Keane, Simon and Solomyak [42] and Ruan, Su and Yao [52]. Here we present a generalised version of fractal interpolation functions G : [0, 1] → R constructed with Markov systems, similar to Deniz and Özdemir [22].…”

“…One way to encode the picture is the fractal image compression method, which concept was first introduced by Barnsley, see [10,11] and Barnsley and Hurd [13] and Barnsley and Elton [12]. Later, the theory developed widely, see for example Fisher [27], Keane, Simon and Solomyak [42], Chung and Hsu [19], Jorgensen and Song [40]. The idea behind the fractal image compression is that a natural image, such as a face, landscape etc., contains a kind of self-similarity.…”