Carnot metrics, dynamics and local rigidity

This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

Information geometry of the space of probability measures and barycenter maps

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures, and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

Comparison of Sr Transport in Compacted Homoionous Na and Ca Bentonite Using a Planar Source Method Evaluated at Ideal and Non-Ideal Boundary Condition

With the aim to determine the influence of dominant interlayer cation on the sorption and diffusion properties of bentonite, diffusion experiments with Sr on the compacted homoionous Ca- and Na-forms of Czech natural Mg/Ca bentonite using the planar source method were performed. The bentonite was compacted to 1400 kg·m−3, and diffusion experiments lasted 1, 3 or 5 days. Two methods of apparent diffusion coefficient Da determination based on the analytical solution of diffusion equation for ideal boundary conditions in a linear form were compared and applied. The determined Da value for Ca-bentonite was 1.36 times higher than that for Na-bentonite sample. Values of Kd were determined in independent batch sorption experiments and were extrapolated for the conditions of compacted bentonite. In spite of this treatment, the use of Kd values determined by batch sorption experiments on a loose material for the determination of effective diffusion coefficient De values from planar source diffusion experiments proved to be inconsistent with the standard Fickian description of diffusion taking into account only the pore diffusion in compacted bentonite. Discrepancies between Kd and De values were measured in independent experiments, and those that resulted from the evaluation of planar source diffusion experiments could be well explained by the phenomenon of surface diffusion. The obtained values of surface diffusion coefficients Ds were similar for both studied systems, and the predicted value of total effective diffusion coefficient De(tot) describing Sr transport in the Na-bentonite was four times higher than in the Ca-bentonite.

Coarse compactifications and controlled products

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

The physical states of a boundary conformal field theory

The path integral of a conformal field theory on a bordered Riemann surface defines a state in a Hilbert space on this boundary. Over the ideal boundary, the Hausdorff dimension may be less than one. The integral representing the flux over the ideal boundary is evaluated through a generalization of the residue theorem. The identification of the state for infinite-genus surfaces with the vacuum state with a perturbative vacuum is distinguished from the Hilbert space on ideal boundaries of nonzero linear measure. This nonperturbative effect is identified as an instanton in a separate quantum theory.