We investigate how the Hausdorff dimension and measure of a self-similar set K aS dagger R (d) behave under linear images. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite, then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under the image of the projection. In general, with no restrictions on T we establish that H (t) (L o O(K)) = H (t) (L(K)) for every element O of the closure of T, where L is a linear map and t = dim (H) K. We also prove that for disjoint subsets A and B of K we have that H (t) (L(A) ai, L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d,R) and the strong separation condition is satisfied, then dimH (g(K)) = min {dim (H) K, l} where g is a continuously differentiable map of rank l. We deduce the same result without any separation condition and we generalize a result of Eroglu by obtaining that H (t) (g(K)) = 0.
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机译：我们研究线性图像下Hausdorff尺寸和自相似集合K aS匕首R（d）的度量。这取决于由K的定义图的正交部分生成的组T的性质。我们证明，如果T是有限的，则K的每个线性图像都是有向图吸引子，并且至少存在K的一个投影尺寸下降到投影图像下方。通常，对于T没有任何限制，我们建立T的每个元素O的H（t）（L o O（K））= H（t）（L（K）），其中L是线性映射并且t =昏暗（H）K。我们还证明，对于K的不相交子集A和B，我们有H（t）（L（A）ai，L（B））=0。Hochman和Shmerkin证明如果T如果在SO（d，R）中具有高密度且满足强分离条件，则dimH（g（K））= min {dim（H）K，l}，其中g是等级l的连续可微图。我们推导了相同的结果，没有任何分离条件，并且通过获得H（t）（g（K））= 0来概括Eroglu的结果。
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