On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
| Autor: | Pablo Shmerkin, Ian Morris |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Lyapunov function
Pure mathematics Applied Mathematics General Mathematics 010102 general mathematics Hausdorff space 01 natural sciences Infimum and supremum 010101 applied mathematics Set (abstract data type) symbols.namesake Dimension (vector space) Hausdorff dimension symbols Affine transformation 0101 mathematics Mathematics |
| Zdroj: | Transactions of the American Mathematical Society. 371:1547-1582 |
| ISSN: | 1088-6850 0002-9947 |
| Popis: | Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Barany, Hochman- Solomyak and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions. |
| Databáze: | OpenAIRE |
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