Existence of full-Hausdorff-dimension invariant measures of dynamical systems with dimension metrics

This paper is concerned with the open problem of the existence of invariant measures of full Hausdorff dimension. Let f : X -> X be a continuous transformation of a compact topological space X. The author introduces a dimension metric to (X, f) and shows that the dynamical system (X, f) has an f-invariant Borel probability measure mu of full Hausdorff dimension, in the sense of a dimension metric d d; namely, HD(d) over cap(mu) = HD(d) over cap(X). If (X, f) is a positively expansive system over a compact metric space X, then there exists a compatible "almost" dimension metric d with respect to which (X, f) has a unique invariant Borel probability measure mu of full Hausdorff dimension.