The goal of this paper is to introduce a new class of transformations of measures on R having the form (heuristically) T = φ · ∇φ/|∇φ|, where φ is some function; our construction exhibits an interesting link between two actively developing areas: optimal transportation and geometrical glows (see [1], [2], and [3] concerning these directions). Let A ⊂ R, d ≥ 2, be a compact convex set and let μ ...

In this study, we prove that a countably infinite number of one-parameterized one-dimensional dynamical systems preserve the Lebesgue measure and are ergodic for measure. The consider connect parameter region in which exact one almost all orbits diverge to infinity correspond critical points weak chaos tends occur (the Lyapunov exponent converging zero). These results generalization work by Adl...

Let A be a bounded subset of R. We give an upper bound on the volume of the symmetric difference of A and f(A) where f is a translation, a rotation, or the composition of both, a rigid motion. The volume is measured by the d-dimensional Hausdorff measure, which coincides with the Lebesgue measure for Lebesgue measurable sets. We bound the volume of the symmetric difference of A and f(A) in term...

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...