A report on an ergodic dichotomy. (English) Zbl 07779085
Author’s abstract: We establish (some directions of) a Ledrappier correspondence between Hölder cocycles, Patterson-Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson-Sullivan measures for \(\vartheta\)-Anosov representations over a local field and show that these are parameterized by the \(\vartheta\)-critical hypersurface of the representation. We use these Patterson-Sullivan measures to establish a dichotomy concerning directions in the interior of the \(\vartheta\)-limit cone of the representation in question: if u is such a half-line, then the subset of u-conical limit points has either total mass if \(|\vartheta| \leq 2\) or zero mass if \(|\vartheta| \geq 4\). The case \(|\vartheta| = 3\) remains unsettled.
Reviewer: Balasubramanian Sury (Bangalore)
Keywords:
Anosov Flows; Patterson-Sullivan measures; Ledrappier correspondence; word-hyperbolic groups; Mineyev flows; Holder cocyclesReferences:
| [1] | Aaronson, J.. An Introduction to Ergodic Theory in Infinite Measure. American Mathematical Society, Providence, RI, 1997. · Zbl 0882.28013 |
| [2] | Akerkar, R.. Nonlinear Functional Analysis. Narosa Publishing House, New Delhi, 1999. · Zbl 0976.46033 |
| [3] | Babillot, M.. Théorie du renouvellement pour des chaînes semi-Markoviennes transientes. Ann. Inst. Henri Poincaré B24(4) (1988), 507-569. · Zbl 0681.60095 |
| [4] | Babillot, M.. Points entiers et groupes discrets: de l’analyse aux systèmes dynamiques. Rigidité, groupe fondamental et dynamique(Panoramas et synthèses, 13). Ed. P. Foulon. Société Mathématique de France, Paris, 2002. · Zbl 1077.11071 |
| [5] | Babillot, M. and Ledrappier, F.. Lalley’s theorem on periodic orbits of hyperbolic flows. Ergod. Th. & Dynam. Sys.18 (1998), 17-39. · Zbl 0915.58074 |
| [6] | Benoist, Y.. Actions propres sur les espaces homogènes réductifs. Ann. of Math. (2)144 (1996), 315-347. · Zbl 0868.22013 |
| [7] | Benoist, Y.. Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal.7(1) (1997), 1-47. · Zbl 0947.22003 |
| [8] | Benoist, Y.. Propriétés asymptotiques des groupes linéaires II. Adv. Stud. Pure Math.26 (2000), 33-48. · Zbl 0960.22012 |
| [9] | Benoist, Y. and Quint, J.-F.. Random Walks on Reductive Groups(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 62), 1st edn. Springer International Publishing, Cham, 2016. · Zbl 1366.60002 |
| [10] | Bochi, J., Potrie, R. and Sambarino, A.. Anosov representations and dominated splittings. J. Eur. Math. Soc. (JEMS)21(11) (2019), 3343-3414. · Zbl 1429.22011 |
| [11] | Bonahon, F.. The geometry of Teichmüller space via geodesic currents. Invent. Math.92 (1988), 139-162. · Zbl 0653.32022 |
| [12] | Borel, A.. Linear Algebraic Groups, 2nd edn.Springer-Verlag, New York, 1991. · Zbl 0726.20030 |
| [13] | Bowditch, B. H.. Convergence groups and configuration spaces. Geometric Group Theory Down Under. Eds. J. Cossey, C. F. Miller , W. D. Neumann and M. Shapiro. de Gruyter, Berlin, 1999, pp. 23-54. · Zbl 0952.20032 |
| [14] | Bowen, R.. Periodic orbits of hyperbolic flows. Amer. J. Math.94 (1972), 1-30. · Zbl 0254.58005 |
| [15] | Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math.95 (1973), 429-460. · Zbl 0282.58009 |
| [16] | Bowen, R. and Ruelle, D.. The ergodic theory of Axiom \(A\) flows. Invent. Math.29 (1975), 181-202. · Zbl 0311.58010 |
| [17] | Bridgeman, M., Canary, R., Labourie, F. and Sambarino, A.. The pressure metric for Anosov representations. Geom. Funct. Anal.25(4) (2015), 1089-1179. · Zbl 1360.37078 |
| [18] | Bridgeman, M., Canary, R., Labourie, F. and Sambarino, A.. Simple roots flows for Hitchin representations. Geom. Dedicata192 (2018), 57-86. William Goldman’s 60th birthday special edition. · Zbl 1383.53038 |
| [19] | Burger, M.. Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2. Int. Math. Res. Not. IMRN7 (1993), 217-225. · Zbl 0829.57023 |
| [20] | Burger, M., Landesberg, O., Lee, M. and Oh, H.. The Hopf-Tsuji-Sullivan dichotomy in higher rank and applications to Anosov subgroups. Preprint, 2022, arXiv:2105.13930. · Zbl 1523.37005 |
| [21] | Canary, R., Zhang, T. and Zimmer, A.. In preparation. |
| [22] | Carvajales, L.. Counting problems for special-orthogonal Anosov representations. Ann. Inst. Fourier (Grenoble)70(3) (2020), 1199-1257. · Zbl 1500.22005 |
| [23] | Carvajales, L.. Growth of quadratic forms under Anosov subgroups. Int. Math. Res. Not. IMRN2023(1) (2023), 785-854. · Zbl 1516.53053 |
| [24] | Carvajales, L., Dai, X., Pozzetti, B. and Wienhard, A.. Thurston’s asymmetric metrics for Anosov representations. Preprint, 2022, arXiv:2210.05292v1. |
| [25] | Chow, M. and Sarkar, P.. Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces. Int. Math. Res. Not. doi: 10.1093/imrn/rnac342. Published online 6 January 2023. · Zbl 1535.37005 |
| [26] | Coelho, Z.. On the asymptotic range of cocyles for shifts of finite type. Ergod. Th. & Dynam. Sys.13 (1993), 249-262. · Zbl 0783.58038 |
| [27] | Constantine, D., Lafont, J.-F. and Thompson, D. J.. Strong symbolic dynamics for geodesic flows on \(\text{CAT}(-1)\) spaces and other metric Anosov flows. J. Éc. polytech. Math.7 (2020), 201-231. · Zbl 1436.37046 |
| [28] | Croke, C. and Fathi, A.. An inequality between energy and intersection. Bull. Lond. Math. Soc.22 (1990), 489-494. · Zbl 0719.53020 |
| [29] | Dey, S. and Kapovich, M.. Patterson-Sullivan theory for Anosov groups. Trans. Amer. Math. Soc.375(12) (2022), 8687-8737. · Zbl 1511.20154 |
| [30] | Ghys, E. and De La Harpe, P. (eds). Sur les groupes hyperboliques d’après Mikhael Gromov. Birkhäuser, Boston, MA, 1990. · Zbl 0731.20025 |
| [31] | Giulietti, P., Kloeckner, B. R., Lopes, A. O. and Marcon, D.. The calculus of thermodynamical formalism. J. Eur. Math. Soc. (JEMS)20(10) (2018), 2357-2412. · Zbl 1410.37031 |
| [32] | Glorieux, O., Monclair, D. and Tholozan, N.. Hausdorff dimension of limit set for projective Anosov groups. Preprint, 2019, arXiv:1902.01844. · Zbl 1527.37025 |
| [33] | Gromov, M.. Hyperbolic groups. Essays in Group Theory. Ed. S. M. Gersten. MSRI Publications, Berkeley, CA,1987, pp. 75-263. · Zbl 0634.20015 |
| [34] | Guéritaud, F., Guichard, O., Kassel, F. and Wienhard, A.. Anosov representations and proper actions. Geom. Topol.21 (2017), 485-584. · Zbl 1373.37095 |
| [35] | Guichard, O. and Wienhard, A.. Anosov representations: domains of discontinuity and applications. Invent. Math.190 (2012), 357-438. · Zbl 1270.20049 |
| [36] | Guivarch’H, Y.. Produits de matrices aléatoires et applications aux propriétés géométriques de sous-groupes du groupe linéaire. Ergod. Th. & Dynam. Sys.10(3) (1990), 483-512. · Zbl 0715.60008 |
| [37] | Helgason, S.. Differential Geometry(Lie Groups and Symmetric Spaces). \( \text{Academic\ Press} \) , New York, 1978. · Zbl 0451.53038 |
| [38] | Hörmander, L.. The Analysis of Partial Differential Operators I(A Series of Comprehensive Studies in Mathematics, 256). Springer-Verlag, Berlin, 1983. · Zbl 0521.35001 |
| [39] | Kaimanovich, V.. Bowen-Margulis and Patterson measures on negatively curved compact manifolds. Dynamical Systems and Related Topics (Nagoya 1990)(Advanced Series in Dynamical Systems, 9). Ed. K. Shiraiwa. World Scientific Publishing, River Edge, NJ, 1991. · Zbl 1529.58002 |
| [40] | Kapovich, M., Leeb, B. and Porti, J.. A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings. Geom. Topol.22 (2018), 3827-3923. · Zbl 1454.53048 |
| [41] | Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995. · Zbl 0878.58020 |
| [42] | Knieper, G.. Volume growth, entropy and the geodesic stretch. Math. Res. Lett.2 (1995), 39-58. · Zbl 0846.53028 |
| [43] | Labourie, F.. Anosov flows, surface groups and curves in projective space. Invent. Math.165 (2006), 51-114. · Zbl 1103.32007 |
| [44] | Ledrappier, F.. Structure au bord des variétés à courbure négative. Sém. Théorie Spect. Géom. Grenoble71 (1994-1995), 97-122. · Zbl 0931.53005 |
| [45] | Ledrappier, F. and Sarig, O.. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete Contin. Dyn. Syst.16(2) (2006), 411-433. · Zbl 1108.37026 |
| [46] | Lee, M. and Oh, H.. Invariant measures for horospherical actions and Anosov groups. Int. Math. Res. Not. doi: 10.1093/imrn/rnac262. Published online 4 October 2022. · Zbl 07794923 |
| [47] | Livšic, A. N.. Cohomology of dynamical systems. Math. USSR Izvestija6 (1972), 1278-1301. · Zbl 0273.58013 |
| [48] | Margulis, G.. Applications of ergodic theory to the investigation of manifolds with negative curvature. Funct. Anal. Appl.3 (1969), 335-336. · Zbl 0207.20305 |
| [49] | Margulis, G.. Certain measures associated with \(\text{U} \) -flows on compact manifolds. Funct. Anal. Appl.4 (1969), 55-67. · Zbl 0245.58003 |
| [50] | Matsumoto, H.. Analyse harmonique dans les systèmes de Tits bornologiques de type affine(Lecture Notes in Mathematics, 590). Springer-Verlag, Berlin, 1977. · Zbl 0366.22001 |
| [51] | Mineyev, I.. Flows and joins of metric spaces. Geom. Topol.9 (2005), 403-482. · Zbl 1137.37314 |
| [52] | Oh, H. and Pan, W.. Local mixing and invariant measures for horospherical subgroups on \(\mathsf{A}\) belian covers. Int. Math. Res. Not. IMRN19 (2019), 6036-6088. · Zbl 1503.37010 |
| [53] | Parry, W. and Pollicott, M.. Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics(Astérisque, 187-188), Société Mathématique de France,Paris, 1990. · Zbl 0726.58003 |
| [54] | Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium States in Negative Curvature(Astérisque, 373). Société Mathématique de France, 2015. · Zbl 1347.37001 |
| [55] | Pollicott, M.. Margulis distribution for Anosov flows. Comm. Math. Phys.113 (1987), 137-154. · Zbl 0633.58037 |
| [56] | Pollicott, M.. Symbolic dynamics for \(S\) male flows. Amer. J. Math.109(1) (1987), 183-200. · Zbl 0628.58042 |
| [57] | Potrie, R. and Sambarino, A.. Eigenvalues and entropy of a \(\mathsf{Hitchin}\) representation. Invent. Math.209 (2017), 885-925. · Zbl 1380.30032 |
| [58] | Pozzetti, B., Sambarino, A. and Wienhard, A.. Anosov representations with Lipschitz limit set. Geom. Topol., to appear, arXiv:1910.06627. |
| [59] | Pozzetti, B., Sambarino, A. and Wienhard, A.. Conformality for a robust class of non conformal attractors. J. Reine Angew. Math.2021(774) (2021), 1-51. · Zbl 1483.37037 |
| [60] | Quint, J.-F.. Cônes limites de sous-groupes discrets des groupes réductifs sur un corps local. Transform. Groups7(3) (2002), 247-266. · Zbl 1018.22004 |
| [61] | Quint, J.-F.. Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv.77 (2002), 563-608. · Zbl 1010.22018 |
| [62] | Quint, J.-F.. Mesures de Patterson-Sullivan en rang supérieur. Geom. Funct. Anal.12 (2002), 776-809. · Zbl 1169.22300 |
| [63] | Ratner, M.. Markov partitions for \(\mathsf{A}\) nosov flows on \(n\) -dimensional manifolds. Israel J. Math.15(1) (1973), 92-114. · Zbl 0269.58010 |
| [64] | Sambarino, A.. Hyperconvex representations and exponential growth. Ergod. Th. & Dynam. Sys.34(3) (2014), 986-1010. · Zbl 1308.37014 |
| [65] | Sambarino, A.. Quantitative properties of convex representations. Comment. Math. Helv.89(2) (2014), 443-488. · Zbl 1295.22016 |
| [66] | Sambarino, A.. The orbital counting problem for hyperconvex representations. Ann. Inst. Fourier (Grenoble)65(4) (2015), 1755-1797. · Zbl 1354.22016 |
| [67] | Schmidt, K.. Cocycles of Ergodic Transformation Groups. Macmillan, India, 1977. · Zbl 0421.28017 |
| [68] | Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci.50 (1979), 171-202. · Zbl 0439.30034 |
| [69] | Tits, J.. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconqe. J. Reine Angew. Math.247 (1971), 196-220. · Zbl 0227.20015 |
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