Equi-homogeneity, Assouad Dimension and Non-autonomous Dynamics

A fractal, as originally described by Mandelbrot, is a set with an irregular and fragmented shape. Many fractals that have been extensively studied, such as self-similar sets, have the same degree of irregularity and fragmentation at all length scales. In contrast to this, an equi-homogeneous set is an irregular and fragmented shape that at each fixed scale is identical at every point. We show that self-similar sets that arise from iterated function systems satisfying the Moran open set condition, canonical examples of ‘fractal sets’, are also equi-homogeneous. Such self-similar sets are also notable in that their Assouad and upper box-counting dimensions are equal. More generally, we show that the Assouad and upper box-counting dimensions are equal for equi-homogeneous sets, provided that the upper and lower box-counting dimensions are equal and are ‘attained’. However, the concept of equi-homogeneity is distinct from any previously defined notion of dimensional equivalence. Using this new theory, we analyse the geometry of a large class of pullback attractors of non-autonomous iterated function systems. These attractors are related to sets that satisfy the Moran structure conditions, include generalized Cantor sets, and possess different dimensional behavior at different length scales. We provide conditions under which the pullback attractor is equi-homogeneous, and use this to compute the Assouad dimension of a certain class of these highly non-trivial sets.