WHEN DOES A BEDFORD–MCMULLEN CARPET HAVE EQUAL HAUSDORFF AND TOPOLOGICAL HAUSDORFF DIMENSIONS?

In this paper, we provide a complete characterization on when the Hausdorff and topological Hausdorff dimension of a given Bedford–McMullen carpet coincide. These two dimensions have a common value if and only if the carpet is either the Cartesian product of [ 0 , 1 ] and a self-similar set, or it is one of the two diagonals of the unit square. [ABSTRACT FROM AUTHOR]