The Hausdorff dimension of integral self-affine sets through fractal perturbations

The estimation or the computation of the Hausdorff dimension of self-affine fractals is of considerable interest. An almost-sure formula for that has been given by K.J. Falconer. However, the precise Hausdorff dimension formulas have been given only in special cases. This problem is even unsolved for a general integral self-affine set F, which is generated by an n x n integer expanding matrix T (not necessarily a similitude) and a finite set A subset of R-n of integer vectors so that F = T-1 (F + A). In this paper, we focus on the pivotal case F subset of R-2 and show that the Hausdorff dimension of F is the limit of a monotonic sequence of McMullen-type dimensions by introducing a process, which we call the fractal perturbation (or deflection) method. In fact, the perturbation method is developed to deal with the pathological case where the characteristic polynomial of T is irreducible over Z. We also consider certain examples of exceptional self-affine fractals for which the Hausdorff dimension is less than the upper bound given by Falconer's formula. These examples show that our approach leads to highly non-trivial computation. (C) 2019 Elsevier Inc. All rights reserved.