The pressure metric for Anosov representations

By M. Bridgeman, Richard Canary, F. Labourie, Andr'es Sambarino. Geometric and Functional Analysis, 2013

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Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out$${(\Gamma)}$$(Γ)-invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group $${\Gamma}$$Γ into $${\mathsf{SL}_m(\mathbb{R})}$$SLm(R). In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil–Petersson metric on the Fuchsian loci. Moreover, we produce $${{\rm Out}(\Gamma)}$$Out(Γ)-invariant metrics on deformation spaces of convex cocompact representations into $${\mathsf{PSL}_2(\mathbb{C})}$$PSL2(C) and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/339d427d7e37c122c81a88575799f73033b209e8

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Citations in the corpus

Invariant Measures for Horospherical Actions and Anosov Groups (2020, background): “9] for general Anosov subgroups in view of the results in [6]:”
Anosov representations and dominated splittings (2016, methodology): “tting Eˆ ‘ is a dominated splitting for the linear bundle automorphism t, with dominating . This is equivalent to the fact that the bundle Hom p ; qis uniformly contracted by the ow induced by t(see [BCLS]). Conversely, dominated splittings for the linear ow tmust be of the form ‘ as above: see Prop”
Anosov groups: local mixing, counting and equidistribution (2020, background): “In [8], the authors defined the geodesic flow associated to ρ which is Hölder equivalent to the Gromov geodesic flow of Σ and showed that this flow is a transitive metric Anosov flow which admits a Markov coding ([6], [7], [39], see also [8, Theorem 3.”
Positivity and higher Teichm\"uller theory (2018, unknown): “→ G/B which sends positive triples in RP1 to positive triples in G/B, where the notion of positivity of triples in G/B is given by Lusztig’s total positivity (see Section 3.1). We refer the reader to [39, 40, 41, 27, 29, 9, 45] for more geometric and dynamical properties of Hitchin representations. ”
Eigenvalues and entropy of a Hitchin representation (2014, background): “ possible that their method also provides a proof. The construction of the ﬂow φi = (φi t: Fiρ → Fi ρ) ∈R is analogous to the construction of the geodesic ﬂow of a projective Anosov representation in [12], this construction is explained in section 3. The advantage of considering this variation is th”
Convex cocompactness in pseudo-Riemannian hyperbolic spaces (2017, background): “[11,12,29,35] to just name a few) and now play a crucial role in higher Teichmüller theory (see e.”
An invitation to higher Teichm\"uller theory (2018, methodology): “A key aspect in the work of Potrie and Sambarino has been the regularity of the map ξ : S1 → G/B of a Hitchin representation.”
Maximally stretched laminations on geometrically finite hyperbolic manifolds (2013, result): “symmetric metric on T (M)Zs. The point of Proposition 1.13 is that dTh(j,ρ) = 0 implies j = ρon T (M)Zs. On the other hand, for convex cocompact M, it follows from work of Burger [Bu, Th.1] (see also [BCLS]) that d′ Th(j,ρ) := log C′(j,ρ) δ(ρ) δ(j) deﬁnes an asymmetric metric on T (M)Zs. Kim [Kim,”
Conformality for a robust class of non-conformal attractors (2019, methodology): “ deﬁned by ρÞÑhλ1 ρ “ limsup tÑ8 1 t log#trγs P rΓs1 : λ1 ` ρpγq ˘ ď tu, is continuous also when K is non-Archimedean. The argument follows the same outline of S. [49] and Bridgeman-Canary-Labourie-S.[8] for K “ R and will actually show that this map is as regular as the association ρÞÑξ1 ρ is. As i”