Related papers: Max-min measures on ultrametric spaces
Consider a measurable space with a finite vector measure. This measure defines a mapping of the $\sigma$-field into a Euclidean space. According to Lyapunov's convexity theorem, the range of this mapping is compact and, if the measure is…
Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an…
The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately…
We study stable conditional measures for a certain equilibrium measure for hyperbolic endomorphisms, on basic sets with overlaps; we show that these conditional measures are geometric probabilities and measures of maximal stable dimension.…
We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set…
We consider the numbers of positive and negative eigenvalues of matrices of squared distances between randomly sampled i.i.d. points in a given metric measure space. These numbers and their limits, as the number of points grows, in fact…
The concentration of measure prenomenon roughly states that, if a set $A$ in a product $\Omega^N$ of probability spaces has measure at least one half, ``most'' of the points of $\Omega^N$ are ``close'' to $A$. We proceed to a systematic…
An ultrametric preserving function $f$ is said to be strongly ultrametric preserving if ultrametrics $d$ and $f \circ d$ define the same topology on $X$ for each ultrametric space $(X,d)$. The set of all strongly ultrametric preserving…
We analyze entropic uncertainty relations in a finite dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the…
We analyze entropic uncertainty relations for two orthogonal measurements on a $N$-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix $U$ relating both bases is distributed according to the Haar…
We consider a group G of isometries acting on a (not necessarily geodesic) delta-hyperbolic space X and possessing a radial limit set of full measure within its limit set. For any continuous quasiconformal measure w supported on the limit…
We consider an isomorphism between the idempotent convexity based on the maximum and the addition operations and the idempotent measure convexity on the maximum and the multiplication operations. We use this isomorphism to investigate…
Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y…
We show that quasi-minimizers of non-homogeneous energy functionals on metric measure spaces are locally H\"older continuous and satisfy the Harnack inequality. We assume that the spaces are doubling and support a Poincar\'e inequality. The…
In this paper, for a metrizable space $Z$, we consider the space of metrics that generate the same topology of $Z$, and that space of metrics is equipped with the supremum metrics. For a metrizable space $X$ and a closed subset $A$ of it,…
In this book, we study Gromov's metric geometric theory on the space of metric measure spaces, based on the idea of concentration of measure phenomenon due to L\'evy and Milman. Although most of the details are omitted in the original…
Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if…
We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional central direction such that the unstable entropy exceeds the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of…
For the functors acting in the category of compact Hausdorff spaces, we introduce the so-called open multi-commutativity property, which generalizes both bicommutativity and openness, and prove that this property is satisfied by the functor…
We introduce and study two new relations between function spaces over measure spaces of infinite measure, motivated by the question of establishing compactness. The first relation captures the uniform decay of function (quasi-)norms ``at…