Equi-homogeneity, Assouad Dimension and Non-autonomous Dynamics

Abstract

We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are 'equi-homogeneous'. This is a regularity property that, roughly speaking, means that at each fixed length-scale any two neighbourhoods of the set have covers of approximately equal cardinality.Self-similar sets are notable in that they are Ahlfors-David regular, which implies that their Assouad and box-counting dimensions coincide. More generally, attractors of non-autonomous iterated functions systems (where maps are allowed to vary between iterations) can have distinct Assouad and box-counting dimensions. Consequently the familiar notion of Ahlfors-David regularity is too strong to be useful in the analysis of this important class of sets, which include generalised Cantor sets and possess different dimensional behaviour at different length-scales. * EJO was partially supported by EPSRC grant EP/G007470/1 while at Warwick on sabbatical leave from University of Nevada Reno.† JCR was partially supported by an EPSRC Leadership Fellowship, grant EP/G007470/1. ‡ NS was partially supported by an EPSRC Career Acceleration Fellowship, grant EP/I004165/1, awarded to Martin Rasmussen, whose support is gratefully acknowledged.We further develop the theory of equi-homogeneity showing that it is a weaker property than Ahlfors-David regularity and distinct from any previously defined notion of dimensional equivalence. However, we show that if the upper and lower box-counting dimensions of an equi-homogeneous set are equal and 'attained' in a sense we make precise then the lower Assouad, Hausdorff, packing, lower box-counting, upper box-counting and Assouad dimensions coincide.Our main results provide conditions under which the attractor of a non-autonomous iterated function system is equi-homogeneous and we use this to compute the Assouad dimension of a certain class of these highly non-trivial sets.

Autonomous choices among deterministic evolution–laws as source of uncertainty

“…Notwithstanding, we can sketch some examples where the insights addressed in this work could already be useful for other disciplines. First, it is interesting to note that (iterated) function systems can be used to study many real life processes like digital communications channels [32], history-dependent neural representation at the Hippocampus [33], turbulent fluid flows [34], actuarial science [35], and even nonunitary quantum dynamics [36]. Also, from the point of view of numerical simulations, it should be worth to explore the use of function systems such as Eq.…”