On self-similar measures with absolutely continuous projections and dimension conservation in each direction

2018/09/26 by Rapaport, Ariel
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.09923

Abstract

Relying on results due to Shmerkin and Solomyak, we show that outside a 0-dimensional set of parameters, for every planar homogeneous self-similar measure ν, with strong separation, dense rotations and dimension greater than 1, there exists q>1 such that \P_zν\_{z∈ S}⊂ L^q(ℝ). Here S is the unit circle and P_zw=⟨ z,w⟩ for w∈ℝ². We then study such measures. For instance, we show that ν is dimension conserving in each direction and that the map z→ P_zν is continuous with respect to the weak topology of L^q(ℝ).

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