102,573 research outputs found
Iterated Function Systems in Mixed Euclidean and p-adic Spaces
We investigate graph-directed iterated function systems in mixed Euclidean
and p-adic spaces. Hausdorff measure and Hausdorff dimension in such spaces are
defined, and an upper bound for the Hausdorff dimension is obtained. The
relation between the Haar measure and the Hausdorff measure is clarified.
Finally, we discus an example in {Bbb R}\times{\Bbb Q}\sb 2 and calculate
upper and lower bounds for its Hausdorff dimension.Comment: 10 pages, 2 Figures; Proceedings of the Conference "Fractal 2006"
held in Vienna, Austria, from February 12 to February 15, 200
Topological diagonalizations and Hausdorff dimension
The Hausdorff dimension of a product XxY can be strictly greater than that of
Y, even when the Hausdorff dimension of X is zero. But when X is countable, the
Hausdorff dimensions of Y and XxY are the same. Diagonalizations of covers
define a natural hierarchy of properties which are weaker than ``being
countable'' and stronger than ``having Hausdorff dimension zero''. Fremlin
asked whether it is enough for X to have the strongest property in this
hierarchy (namely, being a gamma-set) in order to assure that the Hausdorff
dimensions of Y and XxY are the same.
We give a negative answer: Assuming CH, there exists a gamma-set of reals X
and a set of reals Y with Hausdorff dimension zero, such that the Hausdorff
dimension of X+Y (a Lipschitz image of XxY) is maximal, that is, 1. However, we
show that for the notion of a_strong_ gamma-set the answer is positive. Some
related problems remain open.Comment: Small update
Continuous images of Cantor's ternary set
The Hausdorff-Alexandroff Theorem states that any compact metric space is the
continuous image of Cantor's ternary set . It is well known that there are
compact Hausdorff spaces of cardinality equal to that of that are not
continuous images of Cantor's ternary set. On the other hand, every compact
countably infinite Hausdorff space is a continuous image of . Here we
present a compact countably infinite non-Hausdorff space which is not the
continuous image of Cantor's ternary set
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