Hausdorff dimension of planar self-affine sets and measures with overlaps
Michael Hochman, Ariel Rapaport
Abstract
We prove that if \mu is a self-affine measure in the plane whose defining IFS acts totally irreducibly on \mathbb{RP}^1 and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension.We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.
Topics & Concepts
Hausdorff dimensionAffine transformationMathematicsDimension (graph theory)Class (philosophy)Measure (data warehouse)Dimension functionPure mathematicsPlanarHausdorff measureBoundary (topology)Minkowski–Bouligand dimensionCombinatoricsPlane (geometry)Packing dimensionDiscrete mathematicsFractal dimensionMathematical analysisFractalGeometryComputer scienceArtificial intelligenceComputer graphics (images)DatabaseMathematical Dynamics and FractalsAdvanced Topology and Set TheoryAdvanced Differential Equations and Dynamical Systems