# A Weil–Petersson type metric on spaces of metric graphs

@article{Pollicott2014AWT,
title={A Weil–Petersson type metric on spaces of metric graphs},
author={Mark Pollicott and Richard Sharp},
journal={Geometriae Dedicata},
year={2014},
volume={172},
pages={229-244}
}
• Published 4 October 2014
• Mathematics
• Geometriae Dedicata
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## References

SHOWING 1-10 OF 22 REFERENCES
An extension of the Weil–Petersson metric to quasi-Fuchsian space
• Mathematics
• 2006
We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil–Petersson metric on Teichmüller space.
The central limit theorem for geodesic flows onn-dimensional manifolds of negative curvature
AbstractIn this paper we prove a central limit theorem for special flows built over shifts which satisfy a uniform mixing of type $$\gamma ^{n^\alpha }$$ , 00. The function defining the special flow
Moduli of graphs and automorphisms of free groups
• Mathematics
• 1986
This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study
Introduction to Ergodic theory
Hyperbolic dynamics studies the iteration of maps on sets with some type of Lipschitz structure used to measure distance. In a hyperbolic system, some directions are uniformly contracted and others
Thermodynamics, dimension and the Weil–Petersson metric
In this paper we study branched coverings of metrized, simplicial trees F : T → T which arise from polynomial maps f : C → C with disconnected Julia sets. We show that the collection of all such
Introduction to Ergodic Theory
Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time
Outer Space
• K. Vogtmann
• Mathematics
International and Comparative Law Quarterly
• 1975
To investigate the properties of a group G, it is often useful to realize G as a group of symmetries of some geometric object. For example, the classical modular group PSL(2,Z) can be thought of as a
Zeta functions and the periodic orbit structure of hyperbolic dynamics
• Mathematics
• 1990
© Société mathématique de France, 1990, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les
Thurston's Riemannian metric for Teichmüller space
Possibly the most interesting structures on the Teichmuller space are those defined directly from structures on a compact Riemann surface. The quintessential example is due to Riemann himself. The