×

The pressure metric for Anosov representations. (English) Zbl 1360.37078

Summary: 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 \(\mathrm{Out}(\Gamma)\)-invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group \(\Gamma\) into \(\mathrm{SL}_m(\mathbb R)\). 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 \(\mathrm{Out}(\Gamma)\)-invariant metrics on deformation spaces of convex cocompact representations into \(\mathrm{PSL}_2(\mathbb C)\) 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.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
20F65 Geometric group theory
57M50 General geometric structures on low-dimensional manifolds
22E46 Semisimple Lie groups and their representations
58D17 Manifolds of metrics (especially Riemannian)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramov L.M.: “On the entropy of a flow,”. Dokl. Akad. Nauk. SSSR 128, 873-875 (1959) · Zbl 0094.10002
[2] Anderson J.W., Rocha A.: “Analyticity of Hausdorff dimension of limit sets of Kleinian groups,”. Ann. Acad. Sci. Fenn. 22, 349-364 (1997) · Zbl 0890.30028
[3] Benoist Y.: “Propriétés asymptotiques des groupes linéaires,”. Geom. Funct. Anal. 7, 1-47 (1997) · Zbl 0947.22003 · doi:10.1007/PL00001613
[4] Y. Benoist, “Propriétés asymptotiques des groupes linéaires II,” Adv. Stud. Pure Math. 26(2000), 33-48. · Zbl 0960.22012
[5] Y. Benoist, “Automorphismes des cônes convexes,” Invent. Math. 141(2000) 149-193. · Zbl 0957.22008
[6] Y. Benoist, “Convexes divisibles I,” in Algebraic groups and arithmetic, Tata Inst. Fund. Res. Stud. Math. 17(2004), 339-374. · Zbl 1084.37026
[7] Y. Benoist, “Convexes divisibles III,” Ann. Sci. de l’E.N.S. 38(2005), 793-832. · Zbl 1085.22006
[8] L. Bers, “Spaces of Kleinian groups,” in Maryland conference in Several Complex Variables I, Springer-Verlag Lecture Notes in Math, No. 155(1970), 9-34. · Zbl 0211.10602
[9] M. Bridgeman, “Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space,” Geom. and Top. 14(2010), 799-831. · Zbl 1200.30037
[10] M. Bridgeman and E. Taylor, “An extension of the Weil-Petersson metric to quasi-Fuchsian space,” Math. Ann. 341(2008), 927-943. · Zbl 1176.30095
[11] F. Bonahon, “The geometry of Teichmüller space via geodesic currents,” Invent. Math. 92(1988), 139-162. · Zbl 0653.32022
[12] R. Bowen, “Periodic orbits of hyperbolic flows,” Amer. J. Math. 94(1972), 1-30. · Zbl 0254.58005
[13] R. Bowen, “Symbolic dynamics for hyperbolic flows,” Amer. J. Math. 95(1973), 429-460. · Zbl 0282.58009
[14] R. Bowen, “Hausdorff dimension of quasi-circles,” Publ. Math. de l’I.H.E.S. 50(1979), 11-25. · Zbl 0439.30032
[15] R. Bowen and D. Ruelle, “The ergodic theory of axiom A flows,” Invent. Math. 29(1975), 181-202. · Zbl 0311.58010
[16] S. Bradlow, O. Garcia-Prada, and P. Gothen, “Deformations of maximal representations in <InlineEquation ID=”IEq6“> <EquationSource Format=”TEX“>\[{\mathsf{SP}(4,\mathbb{R})} \] <EquationSource Format=”MATHML“> <math xmlns:xlink=”http://www.w3.org/1999/xlink“> <mi mathvariant=”sans-serif“>SP <mo stretchy=”false“>(4, <mi mathvariant=”double-struck“>R <mo stretchy=”false“>),” Quart. J. Math. 63(2012), 795-843. · Zbl 1261.14018
[17] M. Burger, “Intersection, the Manhattan curve and Patterson-Sullivan theory in rank 2,” Internat. Math. Res. Notices7(1993), 217-225. · Zbl 0829.57023
[18] M. Burger, A. Iozzi, F. Labourie, and A. Wienhard, “Maximal representations of surface groups: symplectic Anosov structures,” P.A.M.Q. 1(2005), 555-601. · Zbl 1157.53025
[19] C. Champetier, “Petite simplification dans les groupes hyperboliques,” Ann. Fac. Sci. Toulouse Math. 3(1994), 161-221. · Zbl 0803.53026
[20] D. Cooper and K. Delp, “The marked length spectrum of a projective manifold or orbifold,” Proc. of the A.M.S. 138(2010), 3361-3376. · Zbl 1205.57022
[21] M. Coornaert and A. Papadopoulos, “Symbolic coding for the geodesic flow associated to a word hyperbolic group,” Manu. Math. 109(2002), 465-492. · Zbl 1045.20036
[22] K. Corlette and A. Iozzi, “Limit sets of discrete groups of isometries of exotic hyperbolic spaces,” Trans. A.M.S. 351(1999), 1507-1530. · Zbl 0932.37011
[23] F. Dal’Bo and I. Kim, “A criterion of conjugacy for Zariski dense subgroups,” Comptes Rendus Math. 330 (2000), 647-650. · Zbl 0953.22013
[24] T. Delzant, O. Guichard, F. Labourie, and S. Mozes, “Displacing representations and orbit maps,” in Geometry, rigidity and group actions, Univ. Chicago Press, 2011, 494-514. · Zbl 1281.20047
[25] G. Dreyer, “Length functions for Hitchin representations,” Algebraic and Geometric Topology, to appear, preprint available at: arXiv:1106.6310. · Zbl 1285.57010
[26] M. Darvishzadeh and W. Goldman, “Deformation spaces of convex real projective structures and hyperbolic structures,” J. Kor. Math. Soc. 33(1996), 625-639. · Zbl 0874.53025
[27] M. Gromov, “Hyperbolic groups,” in Essays in Group Theory, MSRI Publ. 8 (1987), 75-263. · Zbl 0634.20015
[28] O. Guichard and A. Wienhard, “Topological Invariants of Anosov representations,” J. Top. 3(2010), 578—642. · Zbl 1225.57012
[29] O. Guichard and A. Wienhard, “Anosov representations: Domains of discontinuity and applications,” Invent. Math. 190(2012), 357-438. · Zbl 1270.20049
[30] O. Guichard, oral communication. · Zbl 0167.50103
[31] H. Gündoğan. “The component group of the automorphism group of a simple Lie algebra and the splitting of the corresponding short exact sequence,” J. Lie Theory20(2010), 709-737. · Zbl 1217.17006
[32] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds Lecture Notes in Mathematics, Vol. 583, 1977 · Zbl 0355.58009
[33] N. Hitchin, “Lie groups and Teichmüller space,” Topology31(1992), 449-473. · Zbl 0769.32008
[34] J. Hubbard, Teichmüler theory and applications to geometry, topology and dynamics. Vol 1., Matrix Editions, Ithaca.
[35] J.E. Humphreys Linear algebraic groups, Graduate Text in Mathematics 21, Springer Verlag , New York, 1981. · Zbl 0890.30028
[36] D. Johnson and J. Millson, “Deformation spaces associated to compact hyperbolic manifolds,” in Discrete Groups and Geometric Analysis, Progress in Math., vol. 67(1987), 48-106. · Zbl 0664.53023
[37] M. Kapovich, Hyperbolic manifolds and discrete groups, Progr. Math. 183, Birkhäuser, 2001. · Zbl 0958.57001
[38] A. Katok, G. Knieper, M. Pollicott, and H. Weiss, “Differentiability and analyticity of topological entropy for Anosov and geodesic flows,” Invent. Math. 98(1989), 581-597. · Zbl 0702.58053
[39] A. Katok, G. Knieper, and H. Weiss, “Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows,” Comm. Math. Phys. 138(1991), 19-31. · Zbl 0749.58041
[40] I. Kim, “Ergodic theory and rigidity on the symmetric space of non-compact type,” Erg. Thy. Dyn. Sys. 21(2001), 93-114. · Zbl 0978.37016
[41] I. Kim, “Rigidity and deformation space of strictly convex real projective structures,” J. Differential Geom. 58(2001), 189-218. · Zbl 1076.53053
[42] I. Kim and G. Zhang, “Kähler metric on Hitchin component”, preprint available at http://arxiv.org/pdf/1312.1965v1.
[43] Koszul J.L.: “Déformation des connexions localement plates,”. Ann. Inst. Four. 18, 103-114 (1968) · Zbl 0167.50103 · doi:10.5802/aif.279
[44] F. Labourie, “Anosov flows, surface groups and curves in projective space,” Invent. Math. 165(2006), 51-114. · Zbl 1103.32007
[45] F. Labourie. “Cross Ratios, Surface Groups, <InlineEquation ID=”IEq7“> <EquationSource Format=”TEX“>\[{\mathsf{SL}_{\rm n}(\mathbb{R})} \] <EquationSource Format=”MATHML“> <math xmlns:xlink=”http://www.w3.org/1999/xlink“> <mi mathvariant=”sans-serif“>SL <mi mathvariant=”normal“>n <mo stretchy=”false“>( <mi mathvariant=”double-struck“>R <mo stretchy=”false“>) and Diffeomorphisms of the Circle,” Publ. Math. de l’I.H.E.S. 106(2007), 139-213. · Zbl 1203.30044
[46] F. Labourie, “Flat projective structures on surfaces and cubic differentials,” P.A.M.Q. 3(2007), 1057-1099. · Zbl 1158.32006
[47] F. Labourie, “Cross ratios, Anosov representations and the energy functional on Teichmüller space,” Ann. Sci. E.N.S. 41(2008), 437-469. · Zbl 1160.37021
[48] F. Labourie “Cyclic surfaces and Hitchin components in rank 2”, preprint available at http://arxiv.org/pdf/1406.4637. · Zbl 1372.32021
[49] F. Labourie, R. Wentworth “The pressure metric along the fuchsian locus” in preparation. · Zbl 1404.37036
[50] F. Labourie, Lectures on representations of surface groups, Zurich Lectures in Advanced Mathematics, 2013, 145 pages. · Zbl 1285.53001
[51] Q. Li, “Teichmüller space is totally geodesic in Goldman space,” preprint, available at: http://front.math.ucdavis.edu/1301.1442 · Zbl 1338.57019
[52] A.N. Livšic, “Cohomology of dynamical systems,” Math. USSR Izvestija6(1972).
[53] J. Loftin, “Affine spheres and convex <InlineEquation ID=”IEq8“> <EquationSource Format=”TEX“>\[{\mathbb{RP}^2} \] <EquationSource Format=”MATHML“> <math xmlns:xlink=”http://www.w3.org/1999/xlink“> <mi mathvariant=”double-struck“>RP2 structures,” Amer. J. Math. 123(2001), 255-274. · Zbl 0997.53010
[54] A. Lubotzky and A. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58(1985), no. 336. · Zbl 0598.14042
[55] R. Mañé, Teoria Ergódica, Projeto Euclides, IMPA, 1983.
[56] C. McMullen, “Thermodynamics, dimension and the Weil-Petersson metric,” Invent. Math. 173(2008), 365-425. · Zbl 1156.30035
[57] I. Mineyev, “Flows and joins of metric spaces,” Geom. Top. 9(2005), 403-482. · Zbl 1137.37314
[58] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque187-188(1990). · Zbl 0726.58003
[59] S. Patterson, “The limit set of a Fuchsian group,” Acta Math. 136(1976), 241-273. · Zbl 0336.30005
[60] M. Pollicott, “Symbolic dynamics for Smale flows,” Am. J. of Math. 109(1987), 183-200. · Zbl 0628.58042
[61] M. Pollicott and R. Sharp, “Length asymptotics in higher Teichmüller theory,” Proc. A.M.S. 142(2014), 101-112. · Zbl 1288.37004
[62] D. Ragozin, “A normal subgroup of a semisimple Lie group is closed.” Proc. A.M.S. 32(1972), 632-633. · Zbl 0231.22012
[63] D. Ruelle, “Repellers for real analytic maps,” Ergodic Theory Dynamical Systems2(1982), 99-107. · Zbl 0506.58024
[64] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, London. · Zbl 0702.58056
[65] A. Sambarino, Quelques aspects des représentations linéaires des groupes hyperboliques, Ph.D. Thesis, Paris Nord, 2011.
[66] A. Sambarino, “Quantitative properties of convex representations,” Comm. Math. Helv., 89(2014), 443-488. · Zbl 1295.22016
[67] A. Sambarino, “Hyperconvex representations and exponential growth,” Ergod. Th. Dynam. Sys. 34(2014), 986-1010. · Zbl 1308.37014
[68] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987. · Zbl 0606.58003
[69] D. Sullivan, “The density at infinity of a discrete group of hyperbolic motions,” Inst. Hautes Études Sci. Publ. Math. 50(1979), 171-202. · Zbl 0439.30034
[70] S. Tapie, “A variation formula for the topological entropy of convex-cocompact manifolds,” Erg. Thy. Dynam. Sys. 31(2011), 1849-1864. · Zbl 1241.53070
[71] Wolpert S.: “Thurston’s Riemannian metric for Teichmüller space,”. J. Diff. Geom. 23, 143-174 (1986) · Zbl 0592.53037
[72] Yue C.: “The ergodic theory of discrete isometry groups on manifolds of variable negative curvature,”. Trans. A.M.S. 348, 4965-5005 (1996) · Zbl 0864.58047 · doi:10.1090/S0002-9947-96-01614-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.