Related papers: Critical exponent for geodesic currents

We define geometric critical exponents for systems that undergo continuous second order classical and quantum phase transitions. These relate scalar quantities on the information theoretic parameter manifolds of such systems, near…

Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the…

We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic…

We endow the space of projective filling geodesic currents on a closed hyperbolic surface with a natural asymmetric metric extending Thurston's asymmetric metric on Teichm\"uller space, as well as analogous metrics arising from Hitchin…

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination…

In this paper we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure theoretic entropy is upper semicontinuous when there is no loss of mass. In case we are losing…

In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which…

We propose a definition for the length of closed geodesics in a globally hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closed geodesics of length less than $R$ grows exponentially fast with $R$…

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves…

Strong hyperbolicity is a coarse notion of negative curvature, stronger than Gromov hyperbolicity, that includes all CAT(-k) metrics for k positive and allows the use of dynamical techniques available in negative curvature, such as…

We work with the space $\mathcal C(S)$ of geodesic currents on a closed surface $S$ of negative Euler characteristic. By prior work of the author with Sebastian Hensel, each filling geodesic current $\mu$ has a unique length-minimizing…

The critical currents in Josephson junctions of conventional superconductors with macroscopic defects are calculated for different defect critical current densities as a function of the magnetic field. We also study the evolution of the…

Within the framework of a generalized Ising model, a one-dimensional magnetic of a finite length with free ends is considered. The correlation length exponent, dynamic critical exponent z of the magnet is calculated taking into account the…

Geodesic currents on closed hyperbolic surfaces are measures on the unit tangent bundle invariant under geodesic flow and orientation reversal. Every geodesic current induces a dual function on curves via the geometric intersection pairing.…

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

Basing upon the recent development of the Patterson-Sullivan measures with a H\"older continuous nonzero potential function, we use tools of both dynamics of geodesic flows and geometric properties of negatively curved manifolds to present…

For metric spaces with curvature less than or equal to x, x<0, it is shown that a recurrent geodesic can be approximated by closed geodesics. A counter example is provided for the converse.

Critical nets in $\mathbb{R}^k$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices…

For every proper geodesic space $X$ we introduce its quasi-geometric boundary $\partial_{QG}X$ with the following properties: 1. Every geodesic ray $g$ in $X$ converges to a point of the boundary $\partial_{QG}X$ and for every point $p$ in…

We introduce a geometry on the cone of positive closed currents of bidegree (p,p) and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like…