Abstract
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 μ of full Hausdorff dimension, in the sense of a dimension metric $$\ifmmode\expandafter\hat\else\expandafter\^\fi{d};$$ namely, $${\text{HD}}_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{d}}} (\mu ) = {\text{HD}}_{{\ifmmode\expandafter\hat\else\expandafter\^\fi{d}}} (X).$$ 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 μ of full Hausdorff dimension.
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