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

Abstract

Relying on results due to Shmerkin and Solomyak, we show that outside a 0-dimensional set of parameters, for every planar homogeneous selfsimilar measure ν, with strong separation, dense rotations and dimension greater than 1, there exists q > 1 such that {Pzν} z∈S ⊂ L q (R). Here S is the unit circle and Pzw = z, w for w ∈ R 2 . We then study such measures. For instance, we show that ν is dimension conserving in each direction and that the map z → Pzν is continuous with respect to the weak topology of L q (R).