On Self-Similar Sets with Overlaps and Inverse Theorems for Entropy in $\mathbb {R}^d$ Susanna C. Calkins and preserving

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On Self-Similar Sets with Overlaps and Inverse Theorems for Entropy in $\mathbb {R}^d$ Susanna C. Calkins and preservingThe author studies self similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $\Phi$ does not preserve a proper affine subspace, he shows that one of the following holds: (1) the dimension is equal to the trivial bound (the minimum of $d$ and the similarity dimension $s$); (2) for all large $n$ there are $n$ fold compositions of maps from $\Phi$ which are super exponentially close in $n$; (3) there is a non