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/PL00001613Search in Google Scholar

[2] Y. Benoist and J.-F. Quint, Random walks on reductive groups, Ergeb. Math. Grenzgeb. (3) 62, Springer, Cham 2016. 10.1007/978-3-319-47721-3Search 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/BF02392456Search 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/905Search in Google Scholar

[5] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1–30. 10.2307/2373590Search 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/BF01389848Search 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/BF02684772Search 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.2006.021Search 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. Search 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. Search 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/s00039-015-0333-8Search 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/s10711-017-0305-2Search 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/S1073792893000236Search in Google Scholar

[14] R. Canary, Hitchin representations of Fuchsian groups, preprint (2021), https://arxiv.org/abs/2110.01043. Search 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.2022.108439Search 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/rnab181Search 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.115Search 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.2914Search 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/BF02803518Search 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. Search 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.1998.037Search 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/s10240-006-0039-4Search 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.2017.21.485Search 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/s00222-012-0382-7Search 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/RM1998v053n02ABEH000017Search 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-ISearch 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/s11856-018-1670-8Search 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.2018.124Search 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/s11856-020-2073-1Search in Google Scholar

[30] M. Kapovich and B. Leeb, Relativizing characterizations of Anosov subgroups. I, preprint (2018), https://arxiv.org/abs/1807.00160. Search 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-ySearch 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.2018.22.3827Search 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.2017016Search 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/s00222-005-0487-3Search 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. 10.1215/00127094-2009-040Search 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/BF02392732Search 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/S0012-7094-92-06810-4Search 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.2008.22.247Search 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/1571882429Search 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.12488Search 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_20Search 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.2707Search in Google Scholar

[43] G. Martone and T. Zhang, Positively ratioed representations, Comment. Math. Helv. 94 (2019), no. 2, 273–345. 10.4171/CMH/461Search 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/S0012-7094-76-04350-7Search 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/CBO9780511543050Search 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. Search in Google Scholar

[47] M. Pollicott, Symbolic dynamics for Smale flows, Amer. J. Math. 109 (1987), no. 1, 183–200. 10.2307/2374558Search 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/S0002-9939-2013-12059-9Search 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/1327Search 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/s00222-017-0721-9Search 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/s00039-002-8266-4Search 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.2017.40Search in Google Scholar

[53] D. Ruelle, Thermodynamic formalism, Encyclopedia Math. Appl. 5, Addison-Wesley, Reading 1978. Search 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.2003.209.365Search in Google Scholar

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

[56] A. Sambarino, Quantitative properties of convex representations, Comment. Math. Helv. 89 (2014), no. 2, 443–488. 10.4171/CMH/324Search 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.2973Search in Google Scholar

[58] A. Sambarino, Infinitesmal Zariski closures of positive representations, preprint (2020), https://arxiv.org/abs/2012.10276. Search 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/S0002-9939-03-06927-2Search 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/089/01485Search in Google Scholar

[61] O. M. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), no. 3, 555–577. 10.1007/s002200100367Search in Google Scholar

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

[63] S. V. Savchenko, Periodic points of countable topological Markov chains, Sb. Math 186 (1995), 1493–1529. 10.1070/SM1995v186n10ABEH000081Search 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/624Search 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.2455Search 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/fm182-3-3Search 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.1240Search in Google Scholar

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

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

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