Date/Time
Date(s) – 25/01/2017
3:00 pm – 4:00 pm
Location
VG02
Categories
Fractal dimension of Moran sets and applications to PDEs
Abstract
The scaling of `local covers’ can vary dramatically between different point of a set. The Assouad dimension is the global upper bound on this scaling. However, even sets that scale identically at every point, such as generalised Cantor sets, can have Assouad dimension strictly greater than their box-counting dimension.
In this talk I will show that a large class of Moran sets are “equi-homogeneous”, which means that the scaling of local covers is uniform across all points of the set. This property can be interpreted as every point of the Moran set having the same amount of local detail. We will see that equi-homogeneity implies that the Assouad dimension can be recovered from the more straightforward box-counting dimensions, provided that the box-counting dimensions are suitably `well behaved’.
Finally, we will look at applications to Partial Differential Equations. For some PDEs, such as reaction-diffusion equations and the Navier-Stokes equations, the eventual behaviour of these infinite-dimensional systems can be described by a finite-dimensional `attracting set’ which all solutions tend towards. This attracting set is a subset of an infinite dimensional space, but can be described by finitely many parameters depending upon its fractal dimension. The Moran sets analysed previously serve as more straightforward prototypes of these important attracting sets.