Abstract. Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface area of the boundaries of parallel sets. In particular, we give characterizations for the Minkowski and the spherical dimensions by means of the Whitney ball count.

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

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.Comment: 12 pages. To appear in DCD