We calculate the Assouad dimension of a planar self-affine set X satisfying the strong separation condition and the projection condition and show that X is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set X adheres to very strong tangential regularity by showing that any two points of X, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.

In this paper, we solve the long standing open problem on exact dimensionality of self-affine measures on the plane. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. In higher dimensions, under certain assumptions, we prove that self-affine and quasi self-affine measures are exact dimensional. In both cases, the measures satisfy the Ledrappier-Young formula.where (p 1 , . . . , p N ) is the associated probability vector.If the linear parts of f i are similarities then we call the self-affine measure self-similar. Ledrappier indicated that applying the method used in [28,29], one could prove exact dimensionality for selfsimilar measures; see Peres and Solomyak [35, p. 1619]. This was later conjectured by Fan, Lau, and Rao [16] and finally confirmed by Feng and Hu [17]. We remark that Feng and Hu [17] proved the result for the push-down measure of any ergodic σ-invariant measure.The first result for self-affine systems is due to McMullen [32], who implicitly proved the exact dimensionality of self-affine measures on the Bedford-McMullen carpets. Later, Gatzouras and Lalley [20] showed the exact dimensionality and calculating the value of dimension of self-affine measures for a class of planar carpet-like self-affine measures. In fact, their method to calculate the Hausdorff dimension of carpet-like self-affine sets was to find the maximal possible dimension of self-affine measures. Later Barański [2] showed similar result for another class of planar selfaffine carpets. In addition to the self-similar case, Feng and Hu [17] proved exact dimensionality for push-down measure of arbitrary ergodic σ-invariant measure on box-like self-affine sets. This