scholarly journals Measured geodesic laminations in Flatland

2014 ◽  
Vol 31 ◽  
pp. 117-136
Author(s):  
Thomas Morzadec
Keyword(s):  
Geodesic Laminations

Geodesic laminations as geometric realizations of Pisot substitutions

2000 ◽  
Vol 20 (4) ◽  
pp. 1253-1266 ◽  
Author(s):  
VÍCTOR F. SIRVENT
Keyword(s):  
Dynamical System ◽  
Geodesic Lamination ◽  
Geodesic Laminations ◽  
Symbolic Dynamical System ◽  
Geometrical Realization

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

scholarly journals Transverse Hölder distributions for geodesic laminations

Topology ◽  
1997 ◽  
Vol 36 (1) ◽  
pp. 103-122 ◽  
Author(s):  
Francis Bonahon
Keyword(s):  
Geodesic Laminations

scholarly journals Products of twists, geodesic lengths and Thurston shears

2014 ◽  
Vol 151 (2) ◽  
pp. 313-350 ◽  
Author(s):  
Scott A. Wolpert
Keyword(s):  
Poisson Bracket ◽  
Symplectic Geometry ◽  
Poisson Algebra ◽  
Weighted Sum ◽  
Hyperbolic Surfaces ◽  
Shear Deformations ◽  
Geodesic Laminations ◽  
Coordinate Algebra ◽  
Left And Right

AbstractThurston introduced shear deformations (cataclysms) on geodesic laminations–deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil–Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary$2$-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.

scholarly journals SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

2020 ◽  
Vol 8 ◽  
Author(s):  
FRANCISCO ARANA-HERRERA ◽  
JAYADEV S. ATHREYA
Keyword(s):  
Moduli Space ◽  
Global Behavior ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Counting Problems ◽  
Finite Area ◽  
Geodesic Laminations ◽  
Square Integrability ◽  
Simple Closed Geodesics ◽  
Hyperbolic Geodesics

Given integers $g,n\geqslant 0$ satisfying $2-2g-n<0$ , let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length ${\leqslant}1$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of ${\mathcal{M}}_{g,n}$ and deduce that $B$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $B$ to statistics of counting problems for simple closed hyperbolic geodesics.

scholarly journals The geometry of measured geodesic laminations and measured train tracks

1989 ◽  
Vol 9 (3) ◽  
pp. 587-604 ◽  
Author(s):  
Howard Weiss
Keyword(s):  
Riemann Surface ◽  
Linear Model ◽  
Tangent Space ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Hölder Continuous ◽  
Geodesic Laminations ◽  
Non Linear ◽  
Train Tracks ◽  
Hyperbolic Riemann Surface

AbstractThurston and Kerckhoff have shown that the space of measured geodesic laminations on a hyperbolic Riemann surface serves as a non-linear model of the tangent space to Teichmüller space at the surface. In this paper we show that the natural map between these manifolds has stronger than Hölder continuous regularity.

Sign in / Sign up

Export Citation Format

Share Document