We prove a that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally, we give a short proof of topological entropy rigidity f...

To each function f of bounded quadratic variation we associate a Hausdorff measure μf . We show that the map f → μf is locally Lipschitz and onto the positive cone of M[0, 1]. We use the measures {μf : f ∈ V2} to determine the structure of the subspaces of V 0 2 which either contain c0 or the square stopping time space S2.

Let E ⊂ C be a compact set, g : C → C be a K-quasiconformal map, and let 0 < t < 2. Let H denote t-dimensional Hausdorff measure. Then H(E) = 0 =⇒ H ′ (gE) = 0 , t′ = 2Kt 2 + (K − 1)t . This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala [2] and answers in the positive a conjecture of K. Astala in op. cit.

Introduction. In this note we study the topological and metric structure of conformal repellor X C K, m > 1, of topological dimension 1. The definition of conformal repellor is given in the next two sections. We then show the following dichotomy: either the Hausdorff dimension of X exceeds 1 or else X is a Jordan curve (simple closed curve) and its 1-dimensional Hausdorff measure is positive an...

Let F : U ⊂ Rn → Rm be a differentiable function and p < m an integer. If k ≥ 1 is an integer, α ∈ [0, 1] and F ∈ Ck+(α), if we set Cp(F ) = {x ∈ U | rank(Df(x)) ≤ p} then the Hausdorff measure of dimension (p + n−p k+α ) of F (Cp(F )) is zero.

We prove that a purely unrectifiable self-similar set of finite 1dimensional Hausdorff measure in the plane, satisfying the Open Set Condition, has radial projection of zero length from every point.

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the setting, this allows us to geometric dimension - akin Hausdorff for metric spaces that distinguishes between e.g. spacelike and null subspaces Minkowski spacetime. The measure corresponding its gives natural reference synthetic or limiting spacetime, what it means such spacetime be collapsed ...

Let Γ ( E ) $\Gamma (E)$ be the family of all paths which meet a set in metric measure space X. The function ↦ A M $E \mapsto AM(\Gamma (E))$ defines $AM$ -modulus X where refers to approximation modulus [22]. We compare $AM(\Gamma Hausdorff c o H 1 $co\mathcal {H}^1(E)$ codimension one and show that ≈ \begin{equation*}\hskip6pc co\mathcal {H}^1(E) \approx (E))\hskip-6pc \end{equation*} for Sus...