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}
}
In this note, we discuss an analogue of the Weil–Petersson metric for spaces of metric graphs and some of its properties. 
Pressure type metrics on spaces of metric graphs
In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil–Petersson metric on the moduli space of metric graphs.
Entropy and the clique polynomial
This paper gives a sharp lower bound on the spectral radius ρ(A) of a reciprocal Perron–Frobenius matrix A∈M2g(Z) , and shows in particular that ρ(A)g⩾(3+5)/2 . This bound supports conjectures on the
Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space
We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection
Central Extension of Mapping Class Group via Chekhov-Fock Quantization
Incompleteness of the pressure metric on the Teichmüller space of a bordered surface
  • Binbin Xu
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2017
We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and that a completion can be given by the moduli space of metrics on a graph (dual to a special ideal
Pressure metrics for deformation spaces of quasifuchsian groups with parabolics
In this paper, we produce a mapping class group invariant pressure metric on the space QF(S) of quasiconformal deformations of a co-finite area Fuchsian group uniformizing a surface S. Our pressure
The geometry of the Weil-Petersson metric in complex dynamics
In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a
An introduction to pressure metrics for higher Teichmüller spaces
We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group
Thermodynamic metrics on outer space
TLDR
The entropy metric and the pressure metric are considered, which can be seen as analogs of the Weil–Petersson metric on the Teichmüller space of a closed surface and it is shown that when the rank r is at least 4, the action of r on the completion of the Culler–Vogtmann outer space using the entropy metric has a fixed point.
Conformally covariant operators and conformal invariants on weighted graphs
Let $$G$$G be a finite connected simple graph. We define the moduli space of conformal structures on $$G$$G. We propose a definition of conformally covariant operators on graphs, motivated by Graham
...
...

References

SHOWING 1-10 OF 22 REFERENCES
An extension of the Weil–Petersson metric to quasi-Fuchsian space
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
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
© 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
...
...