Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

References

[1] Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal. 7 (1997), no. 1, 1–47. 10.1007/PL00001613Suche in Google Scholar

[2] Y. Benoist and J.-F. Quint, Random walks on reductive groups, Ergeb. Math. Grenzgeb. (3) 62, Springer, Cham 2016. 10.1007/978-3-319-47721-3Suche in Google Scholar

[3] C. Bishop and T. Steger, Representation-theoretic rigidity in PSL ⁢ ( 2 , 𝐑 ) , Acta Math. 170 (1993), no. 1, 121–149. 10.1007/BF02392456Suche in Google Scholar

[4] J. Bochi, R. Potrie and A. Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 11, 3343–3414. 10.4171/JEMS/905Suche in Google Scholar

[5] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1–30. 10.2307/2373590Suche in Google Scholar

[6] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. 10.1007/BF01389848Suche in Google Scholar

[7] R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 153–170. 10.1007/BF02684772Suche in Google Scholar

[8] M. Boyle, J. Buzzi and R. Gómez, Almost isomorphism for countable state Markov shifts, J. reine angew. Math. 592 (2006), 23–47. 10.1515/CRELLE.2006.021Suche in Google Scholar

[9] H. Bray, R. Canary and L. Y. Kao, Pressure metrics for deformation spaces of quasifuchsian groups with parabolics, preprint (2020), https://arxiv.org/abs/2006.06184; to apper in Algebr. Geom. Topol. Suche in Google Scholar

[10] H. Bray, R. Canary, L. Y. Kao and G. Martone, Pressure metrics for cusped Hitchin components, preprint (2021), https://arxiv.org/abs/2111.07493. Suche in Google Scholar

[11] M. Bridgeman, R. Canary, F. Labourie and A. Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal. 25 (2015), no. 4, 1089–1179. 10.1007/s00039-015-0333-8Suche in Google Scholar

[12] M. Bridgeman, R. Canary, F. Labourie and A. Sambarino, Simple root flows for Hitchin representations, Geom. Dedicata 192 (2018), 57–86. 10.1007/s10711-017-0305-2Suche in Google Scholar

[13] M. Burger, Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2, Int. Math. Res. Not. IMRN 7 (1993), 217–225. 10.1155/S1073792893000236Suche in Google Scholar

[14] R. Canary, Hitchin representations of Fuchsian groups, preprint (2021), https://arxiv.org/abs/2110.01043. Suche in Google Scholar

[15] R. Canary, T. Zhang and A. Zimmer, Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups, Adv. Math. 404 (2022), Paper No. 108439. 10.1016/j.aim.2022.108439Suche in Google Scholar

[16] L. Carvajales, Growth of quadratic forms under Anosov subgroups, preprint (2020), https://arxiv.org/abs/2004.05903; to appear in Int. Math. Res. Not. IMRN. 10.1093/imrn/rnab181Suche in Google Scholar

[17] D. Constantine, J.-F. Lafont and D. J. Thompson, Strong symbolic dynamics for geodesic flows on CAT ⁢ ( - 1 ) spaces and other metric Anosov flows, J. Éc. polytech. Math. 7 (2020), 201–231. 10.5802/jep.115Suche in Google Scholar

[18] M. Crampon and L. Marquis, Finitude géométrique en géométrie de Hilbert, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 6, 2299–2377. 10.5802/aif.2914Suche in Google Scholar

[19] F. Dal’bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math. 118 (2000), 109–124. 10.1007/BF02803518Suche in Google Scholar

[20] F. Dal’bo and M. Peigné, Comportement asymptotique du nombre de géodésiques fermées sur la surface modulaire en courbure non constante, Études spectrales d’opérateurs de transfert et applications, Astérisque 238, Société Mathématique de France, Paris (1996), 111–177. Suche in Google Scholar

[21] F. Dal’bo and M. Peigné, Some negatively curved manifolds with cusps, mixing and counting, J. reine angew. Math. 497 (1998), 141–169. 10.1515/crll.1998.037Suche in Google Scholar

[22] V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211. 10.1007/s10240-006-0039-4Suche in Google Scholar

[23] F. Guéritaud, O. Guichard, F. Kassel and A. Wienhard, Anosov representations and proper actions, Geom. Topol. 21 (2017), no. 1, 485–584. 10.2140/gt.2017.21.485Suche in Google Scholar

[24] O. Guichard and A. Wienhard, Anosov representations: Domains of discontinuity and applications, Invent. Math. 190 (2012), no. 2, 357–438. 10.1007/s00222-012-0382-7Suche in Google Scholar

[25] B. M. Gurevich and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chains with a countable number of states, Uspekhi Mat. Nauk 53 (1998), no. 2(320), 3–106. 10.1070/RM1998v053n02ABEH000017Suche in Google Scholar

[26] N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449–473. 10.1016/0040-9383(92)90044-ISuche in Google Scholar

[27] G. Iommi, F. Riquelme and A. Velozo, Entropy in the cusp and phase transitions for geodesic flows, Israel J. Math. 225 (2018), no. 2, 609–659. 10.1007/s11856-018-1670-8Suche in Google Scholar

[28] L.-Y. Kao, Manhattan curves for hyperbolic surfaces with cusps, Ergodic Theory Dynam. Systems 40 (2020), no. 7, 1843–1874. 10.1017/etds.2018.124Suche in Google Scholar

[29] L.-Y. Kao, Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces, Israel J. Math. 240 (2020), no. 2, 567–602. 10.1007/s11856-020-2073-1Suche in Google Scholar

[30] M. Kapovich and B. Leeb, Relativizing characterizations of Anosov subgroups. I, preprint (2018), https://arxiv.org/abs/1807.00160. Suche in Google Scholar

[31] M. Kapovich, B. Leeb and J. Porti, Anosov subgroups: Dynamical and geometric characterizations, Eur. J. Math. 3 (2017), no. 4, 808–898. 10.1007/s40879-017-0192-ySuche in Google Scholar

[32] M. Kapovich, B. Leeb and J. Porti, A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings, Geom. Topol. 22 (2018), no. 7, 3827–3923. 10.2140/gt.2018.22.3827Suche in Google Scholar

[33] M. Kesseböhmer and S. Kombrink, A complex Ruelle–Perron–Frobenius theorem for infinite Markov shifts with applications to renewal theory, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 2, 335–352. 10.3934/dcdss.2017016Suche in Google Scholar

[34] F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), no. 1, 51–114.10.1007/s00222-005-0487-3Suche in Google Scholar

[35] F. Labourie and G. McShane, Cross ratios and identities for higher Teichmüller–Thurston theory, Duke Math. J. 149 (2009), no. 2, 279–345. 10.1215/00127094-2009-040Suche in Google Scholar

[36] S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), no. 1–2, 1–55. 10.1007/BF02392732Suche in Google Scholar

[37] S. P. Lalley, Mostow rigidity and the Bishop–Steger dichotomy for surfaces of variable negative curvature, Duke Math. J. 68 (1992), no. 2, 237–269. 10.1215/S0012-7094-92-06810-4Suche in Google Scholar

[38] F. Ledrappier and O. Sarig, Fluctuations of ergodic sums for horocycle flows on ℤ d -covers of finite volume surfaces, Discrete Contin. Dyn. Syst. 22 (2008), no. 1–2, 247–325. 10.3934/dcds.2008.22.247Suche in Google Scholar

[39] J. Loftin, Convex ℝ ⁢ ℙ 2 structures and cubic differentials under neck separation, J. Differential Geom. 113 (2019), no. 2, 315–383. 10.4310/jdg/1571882429Suche in Google Scholar

[40] J. Loftin and T. Zhang, Coordinates on the augmented moduli space of convex ℝ ⁢ ℙ 2 structures, J. Lond. Math. Soc. (2) 104 (2021), no. 4, 1930–1972. 10.1112/jlms.12488Suche in Google Scholar

[41] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math. 123, Birkhäuser, Boston (1994), 531–568. 10.1007/978-1-4612-0261-5_20Suche in Google Scholar

[42] L. Marquis, Surface projective convexe de volume fini, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 1, 325–392. 10.5802/aif.2707Suche in Google Scholar

[43] G. Martone and T. Zhang, Positively ratioed representations, Comment. Math. Helv. 94 (2019), no. 2, 273–345. 10.4171/CMH/461Suche in Google Scholar

[44] H. Masur, Extension of the Weil–Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. 10.1215/S0012-7094-76-04350-7Suche in Google Scholar

[45] R. D. Mauldin and M. Urbański, Graph directed Markov systems, Cambridge Tracts in Math. 148, Cambridge University, Cambridge 2003. 10.1017/CBO9780511543050Suche in Google Scholar

[46] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188, Société Mathématique de France, Paris 1990. Suche in Google Scholar

[47] M. Pollicott, Symbolic dynamics for Smale flows, Amer. J. Math. 109 (1987), no. 1, 183–200. 10.2307/2374558Suche in Google Scholar

[48] M. Pollicott and R. Sharp, Length asymptotics in higher Teichmüller theory, Proc. Amer. Math. Soc. 142 (2014), no. 1, 101–112. 10.1090/S0002-9939-2013-12059-9Suche in Google Scholar

[49] M. Pollicott and M. Urbański, Asymptotic counting in conformal dynamical systems, Mem. Amer. Math. Soc. 271 (2021), no. 1327, 1–139. 10.1090/memo/1327Suche in Google Scholar

[50] R. Potrie and A. Sambarino, Eigenvalues and entropy of a Hitchin representation, Invent. Math. 209 (2017), no. 3, 885–925. 10.1007/s00222-017-0721-9Suche in Google Scholar

[51] J.-F. Quint, Mesures de Patterson–Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002), no. 4, 776–809. 10.1007/s00039-002-8266-4Suche in Google Scholar

[52] F. Riquelme and A. Velozo, Escape of mass and entropy for geodesic flows, Ergodic Theory Dynam. Systems 39 (2019), no. 2, 446–473. 10.1017/etds.2017.40Suche in Google Scholar

[53] D. Ruelle, Thermodynamic formalism, Encyclopedia Math. Appl. 5, Addison-Wesley, Reading 1978. Suche in Google Scholar

[54] S. Ruette, On the Vere–Jones classification and existence of maximal measures for countable topological Markov chains, Pacific J. Math. 209 (2003), no. 2, 366–380. 10.2140/pjm.2003.209.365Suche in Google Scholar

[55] A. Sambarino, Hyperconvex representations and exponential growth, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 986–1010. 10.1017/etds.2012.170Suche in Google Scholar

[56] A. Sambarino, Quantitative properties of convex representations, Comment. Math. Helv. 89 (2014), no. 2, 443–488. 10.4171/CMH/324Suche in Google Scholar

[57] A. Sambarino, The orbital counting problem for hyperconvex representations, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 4, 1755–1797. 10.5802/aif.2973Suche in Google Scholar

[58] A. Sambarino, Infinitesmal Zariski closures of positive representations, preprint (2020), https://arxiv.org/abs/2012.10276. Suche in Google Scholar

[59] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751–1758. 10.1090/S0002-9939-03-06927-2Suche in Google Scholar

[60] O. M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593. 10.1090/pspum/089/01485Suche in Google Scholar

[61] O. M. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), no. 3, 555–577. 10.1007/s002200100367Suche in Google Scholar

[62] O. Sarig, Lecture notes on thermodynamic formalism for topological Markov shifts, 2009. Suche in Google Scholar

[63] S. V. Savchenko, Periodic points of countable topological Markov chains, Sb. Math 186 (1995), 1493–1529. 10.1070/SM1995v186n10ABEH000081Suche in Google Scholar

[64] B. Schapira and S. Tapie, Narrow equidistribution and counting of closed geodesics on noncompact manifolds, Groups Geom. Dyn. 15 (2021), no. 3, 1085–1101. 10.4171/GGD/624Suche in Google Scholar

[65] B. Schapira and S. Tapie, Regularity of entropy, geodesic currents and entropy at infinity, Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 1, 1–68. 10.24033/asens.2455Suche in Google Scholar

[66] M. Stadlbauer, The return sequence of the Bowen-Series map for punctured surfaces, Fund. Math. 182 (2004), no. 3, 221–240. 10.4064/fm182-3-3Suche in Google Scholar

[67] X. Thirion, Groupes de ping-pong et comptage, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), no. 1, 135–190. 10.5802/afst.1240Suche in Google Scholar

[68] K. Tsouvalas, Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps, preprint (2020), https://arxiv.org/abs/2008.04462. Suche in Google Scholar

[69] A. Velozo, Thermodynamic formalism and the entropy at infinity of the geodesic flow, preprint (2017), https://arxiv.org/abs/1711.06796. Suche in Google Scholar

[70] F. Zhu, Ergodicity and equidistribution in Hilbert geometry, preprint (2020), https://arxiv.org/abs/2008.00328. Suche in Google Scholar

[71] F. Zhu, Relatively dominated representations, Ann. Inst. Fourier (Grenoble) 71 (2021), no. 5, 2169–2235. 10.5802/aif.3449Suche in Google Scholar