Antti Käenmäki

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We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the enhancements to the theory required in our work. Our main result is a sharp version of the conical density theorem, which we reduce to a question on rectifiability.
We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery(More)
for μ-almost all points x ∈ R? Here the right-hand sides of (1.1) and (1.2) are denoted by dimloc(μ, x) and dimloc(μ, x), and they are the upper and lower local dimensions of the measure μ at x ∈ R, respectively. We prove that if C is a cone with opening angle at least π, then (1.1) and (1.2) hold for all measures μ and for μ-almost all x ∈ R. Moreover,(More)
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