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Hausdorff Dimensions of Self‐Similar and Self‐Affine Fractals in the Heisenberg Group
Author(s) -
Balogh Zoltán M.,
Tyson Jeremy T.
Publication year - 2005
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504015205
Subject(s) - mathematics , heisenberg group , hausdorff dimension , iterated function system , affine transformation , hausdorff space , affine space , euclidean space , euclidean geometry , invariant (physics) , minkowski–bouligand dimension , effective dimension , pure mathematics , dimension function , fractal , fractal dimension , mathematical analysis , geometry , mathematical physics
We study the Hausdorff dimensions of invariant sets for self‐similar and self‐affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self‐affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra‐Cassano. 2000 Mathematics Subject Classification 22E30, 28A78 (primary), 26A18, 28A78 (secondary).