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May  2016, 36(5): 2711-2727. doi: 10.3934/dcds.2016.36.2711

## Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples

 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  November 2014 Revised  September 2015 Published  October 2015

For conformal graph directed Markov systems, we construct a spectral triple from which one can recover the associated conformal measure via a Dixmier trace. As a particular case, we can recover the Patterson-Sullivan measure for a class of Kleinian groups.
Citation: Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711
##### References:
 [1] R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics,, Bull. Amer. Math. Soc., 25 (1991), 229. doi: 10.1090/S0273-0979-1991-16076-3. Google Scholar [2] S. Albeverio, D. Guido, A. Ponosov and S. Scarlatti, Singular traces and compact operators,, J. Funct. Anal., 137 (1996), 281. doi: 10.1006/jfan.1996.0047. Google Scholar [3] V. Baladi, Positive Transfer Operators and Decay of Correlations,, Advanced Series in Nonlinear Dynamics, (2000). doi: 10.1142/9789812813633. Google Scholar [4] N. Benakli, Polyèdres Hyperboliques Passage du Local au Global,, Thesis, (1992). Google Scholar [5] R. Bhatia and K. Parthasarathy, Lectures on Functional Analysis. Part I. Perturbation by Bounded Operators,, ISI Lecture Notes, (1978). Google Scholar [6] M. Bourdon, Actions Quasi-convexes d'un Groupe Hyperbolique, Flot Géodésique,, Thesis, (1993). Google Scholar [7] R. Bowen, The Hausdorff dimension of quasi-circles,, Publ. Math. IHES, 50 (1979), 11. Google Scholar [8] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Second revised edition, (2008). Google Scholar [9] R. Bowen and C. Series, Markov maps associated with Fuchsian groups,, Publ. Math. IHES, 50 (1979), 153. Google Scholar [10] E. Christensen and C. Ivan, Spectral triples for AF $C^*$-algebras and metrics on the Cantor set,, J. Operator Theory, 56 (2006), 17. Google Scholar [11] A. Connes, Noncommutative Geometry,, Academic Press, (1994). Google Scholar [12] A. Connes, Geometry from the spectral point of view,, Lett. Math. Phys., 34 (1995), 203. doi: 10.1007/BF01872777. Google Scholar [13] J. Conway, A Course in Functional Analysis,, Graduate Texts in Mathematics, (1990). Google Scholar [14] J. Dixmier, Existence de traces non normales,, C. R. Acad. Sci. Paris Sér. A-B, 262 (1966). Google Scholar [15] K. Falconer and T. Samuel, Dixmier traces and coarse multifractal analysis,, Ergodic Theory Dynam. Systems, 31 (2011), 369. doi: 10.1017/S0143385709001102. Google Scholar [16] D. Guido and T. Isola, Fractals in non-commutative geometry,, in Mathematical Physics in Mathematics and Physics, (2000), 171. Google Scholar [17] D. Guido and T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals,, J. Funct. Anal., 203 (2003), 362. doi: 10.1016/S0022-1236(03)00230-1. Google Scholar [18] D. Guido and T. Isola, Dimensions and spectral triples for fractals in $\mathbb R^N$,, in Advances in Operator Algebras and Mathematical Physics, (2005), 89. Google Scholar [19] T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition, (1980). Google Scholar [20] M. Kesseböhmer and T. Samuel, Spectral metric spaces for Gibbs measures,, J. Funct. Anal., 265 (2013), 1801. doi: 10.1016/j.jfa.2013.07.012. Google Scholar [21] M. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums,, Proc. London Math. Soc., 66 (1993), 41. doi: 10.1112/plms/s3-66.1.41. Google Scholar [22] S. Lord, A. Sedaev and F. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators,, J. Funct. Anal., 224 (2005), 72. doi: 10.1016/j.jfa.2005.01.002. Google Scholar [23] R. D. Mauldin and M. Urbanski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,, Cambridge Tracts in Mathematics, (2003). doi: 10.1017/CBO9780511543050. Google Scholar [24] I. Palmer, Riemannian Geometry of Compact Metric Spaces,, Ph.D. Thesis, (2010). Google Scholar [25] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics,, Astérisque, (1990), 1. Google Scholar [26] S. J. Patterson, The limit set of a Fuchsian group,, Acta Math., 136 (1976), 241. doi: 10.1007/BF02392046. Google Scholar [27] J. Pearson and J. Bellissard, Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets,, J. Noncommut. Geom., 3 (2009), 447. doi: 10.4171/JNCG/43. Google Scholar [28] M. Pollicott, A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds,, Amer. J. Math., 117 (1995), 289. doi: 10.2307/2374915. Google Scholar [29] M. Pollicott and R. Sharp, Comparison theorems and orbit counting in hyperbolic geometry,, Trans. Amer. Math. Soc., 350 (1998), 473. doi: 10.1090/S0002-9947-98-01756-5. Google Scholar [30] M. Pollicott and R. Sharp, Poincaré series and comparison theorems for variable negative curvature,, in Topology, (2001), 229. Google Scholar [31] D. Ruelle, Thermodynamic Formalism,, Second edition, (2004). doi: 10.1017/CBO9780511617546. Google Scholar [32] T. Samuel, A Commutative Noncommutative Fractal Geometry,, Ph.D. Thesis, (2010). Google Scholar [33] C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature,, Ergod. Th. and Dynam. Sys., 6 (1986), 601. doi: 10.1017/S0143385700003722. Google Scholar [34] R. Sharp, Periodic orbits of hyperbolic flows,, in On Some Aspects of the Theory of Anosov Systems, (2004), 73. Google Scholar [35] R. Sharp, Spectral triples and Gibbs measures for expanding maps on Cantor sets,, J. Noncommut. Geom., 6 (2012), 801. doi: 10.4171/JNCG/106. Google Scholar [36] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions,, Publ. Math. IHES, 50 (1979), 171. Google Scholar [37] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups,, Acta Math., 153 (1984), 259. doi: 10.1007/BF02392379. Google Scholar [38] P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group,, Acta Math., 152 (1984), 127. doi: 10.1007/BF02392194. Google Scholar [39] J. Várilly, An Introduction to Noncommutative Geometry,, EMS Series of Lectures in Mathematics, (2006). doi: 10.4171/024. Google Scholar [40] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar

show all references

##### References:
 [1] R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics,, Bull. Amer. Math. Soc., 25 (1991), 229. doi: 10.1090/S0273-0979-1991-16076-3. Google Scholar [2] S. Albeverio, D. Guido, A. Ponosov and S. Scarlatti, Singular traces and compact operators,, J. Funct. Anal., 137 (1996), 281. doi: 10.1006/jfan.1996.0047. Google Scholar [3] V. Baladi, Positive Transfer Operators and Decay of Correlations,, Advanced Series in Nonlinear Dynamics, (2000). doi: 10.1142/9789812813633. Google Scholar [4] N. Benakli, Polyèdres Hyperboliques Passage du Local au Global,, Thesis, (1992). Google Scholar [5] R. Bhatia and K. Parthasarathy, Lectures on Functional Analysis. Part I. Perturbation by Bounded Operators,, ISI Lecture Notes, (1978). Google Scholar [6] M. Bourdon, Actions Quasi-convexes d'un Groupe Hyperbolique, Flot Géodésique,, Thesis, (1993). Google Scholar [7] R. Bowen, The Hausdorff dimension of quasi-circles,, Publ. Math. IHES, 50 (1979), 11. Google Scholar [8] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Second revised edition, (2008). Google Scholar [9] R. Bowen and C. Series, Markov maps associated with Fuchsian groups,, Publ. Math. IHES, 50 (1979), 153. Google Scholar [10] E. Christensen and C. Ivan, Spectral triples for AF $C^*$-algebras and metrics on the Cantor set,, J. Operator Theory, 56 (2006), 17. Google Scholar [11] A. Connes, Noncommutative Geometry,, Academic Press, (1994). Google Scholar [12] A. Connes, Geometry from the spectral point of view,, Lett. Math. Phys., 34 (1995), 203. doi: 10.1007/BF01872777. Google Scholar [13] J. Conway, A Course in Functional Analysis,, Graduate Texts in Mathematics, (1990). Google Scholar [14] J. Dixmier, Existence de traces non normales,, C. R. Acad. Sci. Paris Sér. A-B, 262 (1966). Google Scholar [15] K. Falconer and T. Samuel, Dixmier traces and coarse multifractal analysis,, Ergodic Theory Dynam. Systems, 31 (2011), 369. doi: 10.1017/S0143385709001102. Google Scholar [16] D. Guido and T. Isola, Fractals in non-commutative geometry,, in Mathematical Physics in Mathematics and Physics, (2000), 171. Google Scholar [17] D. Guido and T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals,, J. Funct. Anal., 203 (2003), 362. doi: 10.1016/S0022-1236(03)00230-1. Google Scholar [18] D. Guido and T. Isola, Dimensions and spectral triples for fractals in $\mathbb R^N$,, in Advances in Operator Algebras and Mathematical Physics, (2005), 89. Google Scholar [19] T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition, (1980). Google Scholar [20] M. Kesseböhmer and T. Samuel, Spectral metric spaces for Gibbs measures,, J. Funct. Anal., 265 (2013), 1801. doi: 10.1016/j.jfa.2013.07.012. Google Scholar [21] M. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums,, Proc. London Math. Soc., 66 (1993), 41. doi: 10.1112/plms/s3-66.1.41. Google Scholar [22] S. Lord, A. Sedaev and F. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators,, J. Funct. Anal., 224 (2005), 72. doi: 10.1016/j.jfa.2005.01.002. Google Scholar [23] R. D. Mauldin and M. Urbanski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,, Cambridge Tracts in Mathematics, (2003). doi: 10.1017/CBO9780511543050. Google Scholar [24] I. Palmer, Riemannian Geometry of Compact Metric Spaces,, Ph.D. Thesis, (2010). Google Scholar [25] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics,, Astérisque, (1990), 1. Google Scholar [26] S. J. Patterson, The limit set of a Fuchsian group,, Acta Math., 136 (1976), 241. doi: 10.1007/BF02392046. Google Scholar [27] J. Pearson and J. Bellissard, Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets,, J. Noncommut. Geom., 3 (2009), 447. doi: 10.4171/JNCG/43. Google Scholar [28] M. Pollicott, A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds,, Amer. J. Math., 117 (1995), 289. doi: 10.2307/2374915. Google Scholar [29] M. Pollicott and R. Sharp, Comparison theorems and orbit counting in hyperbolic geometry,, Trans. Amer. Math. Soc., 350 (1998), 473. doi: 10.1090/S0002-9947-98-01756-5. Google Scholar [30] M. Pollicott and R. Sharp, Poincaré series and comparison theorems for variable negative curvature,, in Topology, (2001), 229. Google Scholar [31] D. Ruelle, Thermodynamic Formalism,, Second edition, (2004). doi: 10.1017/CBO9780511617546. Google Scholar [32] T. Samuel, A Commutative Noncommutative Fractal Geometry,, Ph.D. Thesis, (2010). Google Scholar [33] C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature,, Ergod. Th. and Dynam. Sys., 6 (1986), 601. doi: 10.1017/S0143385700003722. Google Scholar [34] R. Sharp, Periodic orbits of hyperbolic flows,, in On Some Aspects of the Theory of Anosov Systems, (2004), 73. Google Scholar [35] R. Sharp, Spectral triples and Gibbs measures for expanding maps on Cantor sets,, J. Noncommut. Geom., 6 (2012), 801. doi: 10.4171/JNCG/106. Google Scholar [36] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions,, Publ. Math. IHES, 50 (1979), 171. Google Scholar [37] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups,, Acta Math., 153 (1984), 259. doi: 10.1007/BF02392379. Google Scholar [38] P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group,, Acta Math., 152 (1984), 127. doi: 10.1007/BF02392194. Google Scholar [39] J. Várilly, An Introduction to Noncommutative Geometry,, EMS Series of Lectures in Mathematics, (2006). doi: 10.4171/024. Google Scholar [40] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar
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