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ASSOUAD TYPE DIMENSIONS AND HOMOGENEITY OF FRACTALS
"... Abstract. We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural ‘dimension pair’. In particular, we compute these dimensions for certain classes of selfaffine sets and quasiselfsimilar sets and study their relationships with other notions o ..."
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Abstract. We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural ‘dimension pair’. In particular, we compute these dimensions for certain classes of selfaffine sets and quasiselfsimilar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity. 1.
HAUSDORFF DIMENSION OF WIGGLY METRIC SPACES
"... ABSTRACT. For a compact connected set X ⊆ `∞, we define a quantity β′(x, r) that measures how close X may be approximated in a ball B(x, r) by a geodesic curve. We then show there is c> 0 so that if β′(x, r)> β> 0 for all x ∈ X and r < r0, then dimX> 1+cβ2. This generalizes a theorem ..."
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ABSTRACT. For a compact connected set X ⊆ `∞, we define a quantity β′(x, r) that measures how close X may be approximated in a ball B(x, r) by a geodesic curve. We then show there is c> 0 so that if β′(x, r)> β> 0 for all x ∈ X and r < r0, then dimX> 1+cβ2. This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.