Abstract
Let T1,…,Tm be a family of d×d invertible real matrices with ‖Ti‖<1/2 for 1≤i≤m. For a=(a1,…,am)∈Rmd, let πa:Σ={1,…,m}N→Rd denote the coding map associated with the affine IFS {Tix+ai}i=1m. We show that for every Borel probability measure μ on Σ, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of π⁎aμ is constant for Lmd-a.e. a∈Rmd, where π⁎aμ stands for the push-forward of μ by πa. In particular, we give a necessary and sufficient condition on μ so that π⁎aμ is exact dimensional for Lmd-a.e. a∈Rmd. Moreover, for every analytic set E⊂Σ, each of the Hausdorff, packing, lower and upper box-counting dimensions of πa(E) is constant for Lmd-a.e. a∈Rmd. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.
Published Version
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