arXiv research

Let $X$ be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group $Γ$ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $Γ$-action. Although the construction of Bowen-Margulis m…

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen-Margulis' measure finiteness assumption used in recent work of Ricks is removed. We also construct examples of CAT(0) spaces which do not admit finite Bowen-Margulis measure.

Common perpendiculars equidistribute in negatively curved spaces.

Geodesics in curved spaces spread evenly over time.

Proves finite measure implies product structure for certain discrete subgroups.

Maximal representations show strong entropy rigidity.

Formula for subgroup growth in mapping class groups.

Paper proves finite BMS measure for SPR groups in higher rank Lie groups.

The paper connects geodesic flows and limit sets on visibility manifolds.

The paper describes decompositions of geometric measures on Anosov homogeneous spaces.

New measure of maximal entropy found for a class of geometrically finite groups.

Study counts and equidistributes geodesic orbits on curved spaces.

Frame flows on certain symmetric spaces mix exponentially.

We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $Γ$. If the quotient manifold $\bar X= Γ\backslash X$ is asymptotically $1/4$-pinched, we prove that $Γ$ is divergent and $U\bar X$ has finite Bowen-Margulis measure (which is t…

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C_+, giving error terms and counting with weights, starting from the work of Hu…

The paper studies proper discontinuity of actions on Weyl chamber flow spaces.

The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…

Develops measures for non-Borel Anosov groups on Furstenberg boundary.

Study shows exact dimensionality and regularity of manifolds for specific groups.

We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases: 1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$, 2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$, 3) $G$ is t…

The paper proves rigidity and ergodicity of horospherical foliations.

The paper studies ergodicity of flows on subspaces, generalizing earlier work.

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Ga…

Geometric correspondence links flow metrics to reparameterizations.

Study shows mixing of flows on specific geometric spaces.

Let G:=SO(n,1)^\circ and Γbe a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(Γ\ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and for delta>n-2 for n>= 4, we obtain an {\it effective} archimedean counting result for a dis…

This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every $…

A new flow connects manifold invariants with critical exponents.

Developed a random walk analog of geodesic flow on hyperbolic groups.

Study invariant measures on measured laminations for subgroups of mapping class group.

New set-valued star-shaped risk measures introduced for better risk assessment.

The Bergman measure converges to the Zhang measure on a hybrid space.

Bayesian approach to robust risk measures under model uncertainty.

The paper studies dynamic star-shaped risk measures and their representation.

Introduces Star-Shaped deviation measures for risk analysis.

Transformers can interpolate between arbitrary measures.

Classifies invariant measures on specific character varieties.

Paper characterizes star-shaped risk measures and their properties.

Paper introduces quasi-logconvex risk measures and their properties.

Submodularity is studied for convex risk measures, including Expected Shortfall.

New geometric measure simplifies complex analysis.

The paper explores non-convex risk measures and their characterizations.

The paper calculates extreme measures in continuous time conic finance.

Paper characterizes monotonic mean-deviation risk measures.

One often finds in the literature connections between measures of fairness and measures of feature importance employed to interpret trained classifiers. However, there seems to be no study that compares fairness measures and feature importance measures. In this paper we propose ways to evaluate and compare such measure…