The Divergence of Two Hyperbolic Geodesics

2011 ◽  
Vol 12 (1) ◽  
pp. 101-104 ◽  
Author(s):  
Alan F. Beardon ◽  
David Minda
Keyword(s):  
Hyperbolic Geodesics

scholarly journals Conditioned Brownian motion and hyperbolic geodesics in simply connected domains.

1993 ◽  
Vol 40 (2) ◽  
pp. 321-332 ◽  
Author(s):  
Rodrigo Bañuelos ◽  
Tom Carroll
Keyword(s):  
Brownian Motion ◽  
Conditioned Brownian Motion ◽  
Simply Connected ◽  
Simply Connected Domains ◽  
Hyperbolic Geodesics

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 Carathéodory topology for multiply connected domains I

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mark Comerford
Keyword(s):  
Equivalent Condition ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Hyperbolic Domain ◽  
Carathéodory Topology ◽  
Hyperbolic Geodesics ◽  
Hyperbolic Domains

AbstractWe consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.

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