Analysis of invariant measures in dynamical systems by Hausdorff measure (Q1077568)

Hausdorff measure is a preliminary concept in the definition of Hausdorff dimension, which is one concept of the degree of singularity of a finite measure. In general, Hausdorff measure does not permit as detailed an analysis of an arbitrary natural invariant measure arising from a dynamical system as Lebesgue measure permits of an absolutely continuous measure. It is shown that even for a dynamical system as simple as a modified Baker's transformation, the natural invariant measure has no representation as an indefinite integral with respect to any Hausdorff measure. However, Hausdorff measure can be used to compare different natural invariant measures according to degree of singularity even when their Hausdorff dimensions are identical.