The Dimension Theory of Almost Self-affine Sets and Measures

Abstract

A self-affine IFS \(\mathcal{F} = \left \{f_{i}(x) = A_{i}x + t_{i}\right \}_{i=1}^{m}\) is a finite list of contracting affine maps on \(\mathbb{R}^{d}\), for some d ≥ 1. The attractor of \(\mathcal{F}\) is $$\displaystyle{ \Lambda =\bigcap _{ n=1}^{\infty }\bigcap _{ i_{1},\ldots,i_{n}}f_{i_{1}} \circ \cdots \circ f_{i_{n}}(B), }$$ (0.1) where B is a sufficiently large ball centered at the origin. In most cases we cannot compute the dimension of \(\Lambda \). However, if we add an independent additive random error to each \(f_{i_{k}}\) in (0.1) then the dimension of this random perturbation (called almost self-affine system) is almost surely the so-called affinity dimension of the original deterministic system. The dimension theory of almost self-affine sets and measures were described in Jordan et al. (Commun. Math. Phys. 270(2):519–544, 2007). The multifractal analysis of almost self-affine measures has been studied in some recent papers (Falconer, Nonlinearity 23:1047–1069, 2010; Barral and Feng, Commun. Math. Phys. 318(2):473–504, 2013). In the second part of this note I give a survey of this field but first we review some results related to the dimension theory of self-affine sets.KeywordsRandom fractalsHausdorff dimensionProcesses in random environment2000 Mathematics Subject Classification.37H1537C45