Ergodic fractal measures and dimension conservation

A linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For homogeneous fractals (to be defined), there is a phenomenon of dimension conservation. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This almost everywhere result implies a non-probabilistic statement for homogeneous fractals.