On the dimensions of certain incommensurably constructed sets.
On the double -integral.
On the duality between -modulus and probability measures
Motivated by recent developments on calculus in metric measure spaces , we prove a general duality principle between Fuglede’s notion [15] of -modulus for families of finite Borel measures in and probability measures with barycenter in , with dual exponent of . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in . In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence...
On the dynamical meaning of spectral dimensions
On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation
On the equivalence between CH and the existence of certain -Luzin subsets of .
On the Euler Characteristic in Spaces with a Separability Property.
On the existence of conditional density functions
On the existence of disintegrations
On the existence of measures on σ-algebras
On the Existence of Pathological Submeasures and the Construction of Exotic Topological Groups.
On the Extension of a Measure on Lattices
On the extension of measures.
We give necessary and sufficient conditions for a totally ordered by extension family (Ω, Σx, μx)x ∈ X of spaces of probability to have a measure μ which is an extension of all the measures μx. As an application we study when a probability measure on Ω has an extension defined on all the subsets of Ω.
On the Extension of Measures in Relatively Complemented Lattices
On the Extremality of measure extensions.
On the extremality of regular extensions of contents and measures
Let be an algebra and a lattice of subsets of a set . We show that every content on that can be approximated by in the sense of Marczewski has an extremal extension to a -regular content on the algebra generated by and . Under an additional assumption, we can also prove the existence of extremal regular measure extensions.
On the Functional Equation f(x)g(y) = ...hi(aix + biy).
On the geometric structure of the limit set of conformal iterated function systems.
On the Hausdorff dimension of a family of self-similar sets with complicated overlaps
We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension...
On the Hausdorff Dimension of CAT(κ) Surfaces
We prove 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 for geodesic flow on certain CAT(−1) manifolds.