Chapter 6 Further Research
6.2 Asymptotic statistics
One goal of further research is to remove (A2) from the list of assumptions in Theorems 5.1.1 and 5.1.3. If it is possible to prove the asymptotic
lim m→∞ 1 #Ck+2m X g∈Ck Θ(x) k+ 2m = Z f dµ0
without (A2) then equidistribution with respect to µ0, the measure of maximal entropy, holds for each non-trivial conjugacy class inF.
We have considered statistical properties for free groups acting convex co- compactly on hyperbolic space. It is worth investigating whether these statistical properties hold in the more general setting of negatively curved convex co-compact surfaces. We first give some historical context for the problem.
Let M be a compact surface with constant negative curvature and gt :
T1M → T1M the geodesic flow. Let λ(γ) be the least period of a closed orbit γ of the flow and byλf(γ) the integral of a H¨older continuous functionf :T1M →Ron γ so that
λf(γ) =
Z λ(γ)
0
f(gtx)dt
forx∈γ. Bowen [3] showed that the closed orbits of the geodesic flow are equidis- tributed according to µ the measure of maximal entropy. By equidistribution, we mean the following limit holds
lim T→∞ P {γ:λ(γ)≤T}λf(γ) P {γ:λ(γ)≤T}λ(γ) → Z f dµ.
Since each conjugacy classes in π1(M, x) has an associated closed geodesic. Thus when we consider asymptotics in conjugacy classes we instead study based geodesic loops in the manifold.
LetM be a negatively curved convex co-compact surface (or, again, a man- ifold if there are closed geodesics whose lengths have irrational ratio) whose funda- mental groups are free. For a given basepointx∈M, letCbe a non-trivial conjugacy class inπ1(M, x) and consider the set of geodesic loopsγg based atxfor eachg∈C.
We ask whether this restricted set of geodesics is asymptotically equidistributed with respect to the measure of maximal entropy projected toM. We can expect to use a symbolic approximation technique developed in [50] for orbit counting in the context of hyperbolic flows on compact manifolds. This approximation was adapted in [51] for orbit counting and, importantly, a refined directional asymptotic in the harder setting ofZkcovers of compact negatively curved manifolds. Here the results
concern orbit segments of the geodesic flow onT1M, but the asymptotic statements push down to geodesic loops in M. A potential next step is to consider compact negatively curved surfaces.
Bibliography
[1] A. Beardon. The geometry of discrete groups, volume 91 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1983.
[2] M. Bourdon. Actions quasi-convexes d’un group hyperbolique, flot g´eod´esique. PhD thesis, Orsay, 1993.
[3] R. Bowen. Periodic orbits for hyperbolic flows. Amer. J. Math., 94:1–30, 1972. [4] R. Bowen and C. Series. Markov maps associated with Fuchsian groups. Inst.
Hautes ´Etudes Sci. Publ. Math., 50:153–170, 1979.
[5] A. Broise-Alamichel, J. Parkkonen, and F. Paulin. Equidistribution and count- ing under equilibrium states in negatively curved spaces and graphs of groups. Applications to non-Archimedean Diophantine approximation. ArXiv e-prints, 2016. Preprint arXiv:1612.06717.
[6] R. Brooks. Some remarks on bounded cohomology. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ.
New York, Stony Brook, N.Y., 1978), volume 97 ofAnn. of Math. Stud., pages
53–63. Princeton Univ. Press, Princeton, N.J., 1981.
[7] D. Calegari. The ergodic theory of hyperbolic groups. InGeometry and topology
down under, volume 597 of Contemp. Math., pages 15–52. Amer. Math. Soc.,
Providence, RI, 2013.
[8] D. Calegari and K. Fujiwara. Combable functions, quasimorphisms, and the central limit theorem.Ergodic Theory Dynam. Systems, 30(5):1343–1369, 2010.
[9] Z. Coelho and W. Parry. Central limit asymptotics for shifts of finite type. Israel J. Math., 69(2):235–249, 1990.
[10] M. Coornaert. Mesures de Patterson-Sullivan sur le bord d’un espace hyper- bolique au sens de Gromov. Pacific J. Math., 159(2):241–270, 1993.
[11] F. Dal’bo. Remarques sur le spectre des longueurs d’une surface et comptages. Bol. Soc. Brasil. Mat. (N.S.), 30(2):199–221, 1999.
[12] J. Delsarte. Sur le gitter fuchsien. C. R. Acad. Sci. Paris, 214:147–149, 1942. [13] D. Dolgopyat. Prevalence of rapid mixing in hyperbolic flows. Ergodic Theory
Dynam. Systems, 18(5):1097–1114, 1998.
[14] F. Douma. A lattice point problem on the regular tree. Discrete Math., 311(4):276–281, 2011.
[15] W. Ellison and F. Ellison. Prime numbers. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York; Hermann, Paris, 1985.
[16] W. Feller. An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney, 1971. [17] F. Gantmacher. The Theory of Matrices Vol. II. AMS Chelsea Publishing
Series. Amer. Math. Soc., 2000.
[18] ´E. Ghys and P. de la Harpe. Quasi-isom´etries et quasi-g´eod´esiques. In Sur
les groupes hyperboliques d’apr`es Mikhael Gromov (Bern, 1988), volume 83 of
Progr. Math., pages 79–102. Birkh¨auser Boston, Boston, MA, 1990.
[19] M. Gromov. Hyperbolic groups. InEssays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75–263. Springer, New York, 1987.
[20] L. Guillop´e. Entropies et spectres. Osaka J. Math., 31(2):247–289, 1994. [21] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.
[22] D. Hejhal. The Selberg trace formula for PSL(2,R). Vol. I. Lecture Notes in Mathematics, Vol. 548. Springer-Verlag, Berlin-New York, 1976.
[23] M. Horsham. Central limit theorems for quasi-morphisms of surface groups. PhD thesis, Manchester, 2008.
[24] M. Horsham and R. Sharp. Lengths, quasi-morphisms and statistics for free groups. In Spectral analysis in geometry and number theory, volume 484 of Contemp. Math., pages 219–237. Amer. Math. Soc., Providence, RI, 2009. [25] H. Huber. ¨Uber eine neue Klasse automorpher Funktionen und ein Gitterpunk-
tproblem in der hyperbolischen Ebene. I. Comment. Math. Helv., 30:20–62, 1956.
[26] H. Huber. Zur analytischen Theorie hyperbolischen Raumformen und Bewe- gungsgruppen. Math. Ann., 138:1–26, 1959.
[27] H. Huber. Zur analytischen Theorie hyperbolischer Raumformen und Bewe- gungsgruppen. II. Math. Ann., 142:385–398, 1961.
[28] H. Huber. Zur analytischen Theorie hyperbolischer Raumformen und Bewe- gungsgruppen. II. Nachtrag zu Math. Ann. 142:385–398, 1961. Math. Ann., 143:463–464, 1961.
[29] A. Ingham. The distribution of prime numbers. Cambridge Mathematical Li- brary. Cambridge University Press, Cambridge, 1990.
[30] I. Kapovich and N. Benakli. Boundaries of hyperbolic groups. InCombinatorial
and geometric group theory, volume 296 ofContemp. Math., pages 39–93. Amer.
Math. Soc., Providence, RI, 2002.
[31] I. Kapovich and T. Nagnibeda. The Patterson–Sullivan embedding and minimal volume entropy for outer space. Geom. Funct. Anal., 17(4):1201–1236, 2007. [32] T. Kato. Perturbation theory for linear operators. Classics in Mathematics.
[33] G. Kenison and R. Sharp. Orbit counting in conjugacy classes for free groups acting on trees. J. Topol. Anal., 9(4):631–647, 2017.
[34] Y. Kifer. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc., 321(2):505–524, 1990.
[35] S. Krantz. Function theory of several complex variables. AMS Chelsea Publish- ing, Providence, RI, 2001. Reprint of the 1992 edition.
[36] S. Lalley. Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits. Acta Math., 163(1):1– 55, 1989.
[37] P. Lax and R. Phillips. The asymptotic distribution of lattice points in Eu- clidean and non-Euclidean spaces. J. Funct. Anal., 46(3):280–350, 1982. [38] R. Lyndon. Length functions in groups. Math. Scand., 12:209–234, 1963. [39] G. Margulis. On some aspects of the theory of Anosov systems. Springer
Monographs in Mathematics. Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska.
[40] P. Nicholls. A lattice point problem in hyperbolic space. Michigan Math. J., 30(3):273–287, 1983.
[41] P. Nicholls. The Ergodic Theory of Discrete Groups. London Mathematical Society Lecture Note Series. Cambridge University Press, 1989.
[42] J. Parkkonen and F. Paulin. On the hyperbolic orbital counting problem in conjugacy classes. Math. Z., 279(3):1175–1196, 2015.
[43] W. Parry. An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Israel J. Math., 45(1):41–52, 1983.
[44] W. Parry and M. Pollicott. Zeta functions and the periodic orbit structure
of hyperbolic dynamics. Ast´erisque. 187-188. Paris: Soci´et´e Math´ematique de
France, 1990.
[45] S. J. Patterson. A lattice-point problem in hyperbolic space. Mathematika, 22(1):81–88, 1975.
[46] S. J. Patterson. Corrigendum to: “A lattice-point problem in hyperbolic space” (Mathematika 22(1975), no. 1, 81–88). Mathematika, 23(2):227, 1976.
[47] S. J. Patterson. The limit set of a Fuchsian group. Acta Math., 136(3-4):241– 273, 1976.
[48] S. J. Patterson. On a lattice-point problem in hyperbolic space and related questions in spectral theory. Ark. Mat., 26(1):167–172, 1988.
[49] M Pollicott. Meromorphic extensions of generalised zeta functions. Invent. Math., 85(1):147–164, 1986.
[50] M. Pollicott. A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds. Amer. J. Math., 117(2):289–305, 1995.
[51] M. Pollicott and R. Sharp. Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature.Invent. Math., 117(2):275– 302, 1994.
[52] M. Pollicott and R. Sharp. Large deviations and the distribution of pre-images of rational maps. Comm. Math. Phys., 181(3):733–739, 1996.
[53] M. Pollicott and R. Sharp. The circle problem on surfaces of variable negative curvature. Monatsh. Math., 123(1):61–70, 1997.
[54] M. Pollicott and R. Sharp. Comparison theorems and orbit counting in hyper- bolic geometry. Trans. Amer. Math. Soc., 350(2):473–499, 1998.
[55] M. Pollicott and R. Sharp. Error terms for closed orbits of hyperbolic flows. Ergodic Theory Dynam. Systems, 21(2):545–562, 2001.
[56] M. Pollicott and R. Sharp. Poincar´e series and comparison theorems for variable negative curvature. InTopology, ergodic theory, real algebraic geometry, volume 202 of Amer. Math. Soc. Transl. Ser. 2, pages 229–240. Amer. Math. Soc., Providence, RI, 2001.
[57] M. Pollicott and R. Sharp. Statistics of matrix products in hyperbolic geome- try. In Dynamical numbers—interplay between dynamical systems and number
theory, volume 532 ofContemp. Math., pages 213–230. Amer. Math. Soc., Prov-
idence, RI, 2010.
[58] I Rivin. Growth in free groups (and other stories)—twelve years later. Illinois J. Math., 54(1):327–370, 2010.
[59] D. Ruelle. Statistical mechanics of a one-dimensional lattice gas.Comm. Math. Phys., 9:267–278, 1968.
[60] D. Ruelle. Thermodynamic formalism, volume 5 of Encyclopedia of Mathe- matics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., 1978.
[61] A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.), 20:47–87, 1956.
[62] C. Series. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergodic Theory Dynam. Systems, 6:601–625, 1986.
[63] R. Sharp. Sector estimates for Kleinian groups. Port. Math., 58(4):461–472, 2001.
[64] R. Sharp. Distortion and entropy for automorphisms of free groups. Discrete Contin. Dynam. Systems, 26(1):347–363, 2009.
[65] R. Sharp. Comparing length functions on free groups. In Spectrum and Dy-
namics: Proceedings of the Workshop Held in Montr´eal, QC, April 7-11, 2008:
[66] D. Sullivan. The density at infinity of a discrete group of hyperbolic motions.
Inst. Hautes ´Etudes Sci. Publ. Math., 50:171–202, 1979.
[67] P. Walters. An introduction to ergodic theory, volume 79 ofGraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1982.