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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 and Continuous Dynamical Systems, 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), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.

[2]

S. Albeverio, D. Guido, A. Ponosov and S. Scarlatti, Singular traces and compact operators, J. Funct. Anal., 137 (1996), 281-302. doi: 10.1006/jfan.1996.0047.

[3]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[4]

N. Benakli, Polyèdres Hyperboliques Passage du Local au Global, Thesis, Paris Sud, 1992.

[5]

R. Bhatia and K. Parthasarathy, Lectures on Functional Analysis. Part I. Perturbation by Bounded Operators, ISI Lecture Notes, 3, Macmillan Co. of India, Ltd., New Delhi, 1978.

[6]

M. Bourdon, Actions Quasi-convexes d'un Groupe Hyperbolique, Flot Géodésique, Thesis, Paris Sud, 1993.

[7]

R. Bowen, The Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1979), 11-25.

[8]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, with a preface by David Ruelle, edited by Jean-René Chazottes, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.

[9]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math. IHES, 50 (1979), 153-170.

[10]

E. Christensen and C. Ivan, Spectral triples for AF $C^*$-algebras and metrics on the Cantor set, J. Operator Theory, 56 (2006), 17-46.

[11]

A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.

[12]

A. Connes, Geometry from the spectral point of view, Lett. Math. Phys., 34 (1995), 203-238. doi: 10.1007/BF01872777.

[13]

J. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1990.

[14]

J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B, 262 (1966), A1107-A1108.

[15]

K. Falconer and T. Samuel, Dixmier traces and coarse multifractal analysis, Ergodic Theory Dynam. Systems, 31 (2011), 369-381. doi: 10.1017/S0143385709001102.

[16]

D. Guido and T. Isola, Fractals in non-commutative geometry, in Mathematical Physics in Mathematics and Physics, Sienna 2000, Field Institute Communications, 30, American Mathematical Society, Providence, RI, 2001, 171-186.

[17]

D. Guido and T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal., 203 (2003), 362-400. doi: 10.1016/S0022-1236(03)00230-1.

[18]

D. Guido and T. Isola, Dimensions and spectral triples for fractals in $\mathbb R^N$, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., 5, Theta, Bucharest, 2005, 89-108.

[19]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[20]

M. Kesseböhmer and T. Samuel, Spectral metric spaces for Gibbs measures, J. Funct. Anal., 265 (2013), 1801-1828. doi: 10.1016/j.jfa.2013.07.012.

[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), 41-69. doi: 10.1112/plms/s3-66.1.41.

[22]

S. Lord, A. Sedaev and F. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal., 224 (2005), 72-106. doi: 10.1016/j.jfa.2005.01.002.

[23]

R. D. Mauldin and M. Urbanski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543050.

[24]

I. Palmer, Riemannian Geometry of Compact Metric Spaces, Ph.D. Thesis, Georgia Tech, 2010.

[25]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 1-268.

[26]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.

[27]

J. Pearson and J. Bellissard, Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets, J. Noncommut. Geom., 3 (2009), 447-480. doi: 10.4171/JNCG/43.

[28]

M. Pollicott, A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds, Amer. J. Math., 117 (1995), 289-305. doi: 10.2307/2374915.

[29]

M. Pollicott and R. Sharp, Comparison theorems and orbit counting in hyperbolic geometry, Trans. Amer. Math. Soc., 350 (1998), 473-499. doi: 10.1090/S0002-9947-98-01756-5.

[30]

M. Pollicott and R. Sharp, Poincaré series and comparison theorems for variable negative curvature, in Topology, Ergodic Theory, Real Algebraic Geometry, Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, 2001, 229-240.

[31]

D. Ruelle, Thermodynamic Formalism, Second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617546.

[32]

T. Samuel, A Commutative Noncommutative Fractal Geometry, Ph.D. Thesis, St. Andrews University, 2010.

[33]

C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergod. Th. and Dynam. Sys., 6 (1986), 601-625. doi: 10.1017/S0143385700003722.

[34]

R. Sharp, Periodic orbits of hyperbolic flows, in On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004, 73-138.

[35]

R. Sharp, Spectral triples and Gibbs measures for expanding maps on Cantor sets, J. Noncommut. Geom., 6 (2012), 801-817. doi: 10.4171/JNCG/106.

[36]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES, 50 (1979), 171-202.

[37]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379.

[38]

P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math., 152 (1984), 127-140. doi: 10.1007/BF02392194.

[39]

J. Várilly, An Introduction to Noncommutative Geometry, EMS Series of Lectures in Mathematics, European Mathematical Society, Zürich, 2006. doi: 10.4171/024.

[40]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

show all references

References:
[1]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc., 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.

[2]

S. Albeverio, D. Guido, A. Ponosov and S. Scarlatti, Singular traces and compact operators, J. Funct. Anal., 137 (1996), 281-302. doi: 10.1006/jfan.1996.0047.

[3]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[4]

N. Benakli, Polyèdres Hyperboliques Passage du Local au Global, Thesis, Paris Sud, 1992.

[5]

R. Bhatia and K. Parthasarathy, Lectures on Functional Analysis. Part I. Perturbation by Bounded Operators, ISI Lecture Notes, 3, Macmillan Co. of India, Ltd., New Delhi, 1978.

[6]

M. Bourdon, Actions Quasi-convexes d'un Groupe Hyperbolique, Flot Géodésique, Thesis, Paris Sud, 1993.

[7]

R. Bowen, The Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1979), 11-25.

[8]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, with a preface by David Ruelle, edited by Jean-René Chazottes, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.

[9]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math. IHES, 50 (1979), 153-170.

[10]

E. Christensen and C. Ivan, Spectral triples for AF $C^*$-algebras and metrics on the Cantor set, J. Operator Theory, 56 (2006), 17-46.

[11]

A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.

[12]

A. Connes, Geometry from the spectral point of view, Lett. Math. Phys., 34 (1995), 203-238. doi: 10.1007/BF01872777.

[13]

J. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1990.

[14]

J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B, 262 (1966), A1107-A1108.

[15]

K. Falconer and T. Samuel, Dixmier traces and coarse multifractal analysis, Ergodic Theory Dynam. Systems, 31 (2011), 369-381. doi: 10.1017/S0143385709001102.

[16]

D. Guido and T. Isola, Fractals in non-commutative geometry, in Mathematical Physics in Mathematics and Physics, Sienna 2000, Field Institute Communications, 30, American Mathematical Society, Providence, RI, 2001, 171-186.

[17]

D. Guido and T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal., 203 (2003), 362-400. doi: 10.1016/S0022-1236(03)00230-1.

[18]

D. Guido and T. Isola, Dimensions and spectral triples for fractals in $\mathbb R^N$, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., 5, Theta, Bucharest, 2005, 89-108.

[19]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[20]

M. Kesseböhmer and T. Samuel, Spectral metric spaces for Gibbs measures, J. Funct. Anal., 265 (2013), 1801-1828. doi: 10.1016/j.jfa.2013.07.012.

[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), 41-69. doi: 10.1112/plms/s3-66.1.41.

[22]

S. Lord, A. Sedaev and F. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal., 224 (2005), 72-106. doi: 10.1016/j.jfa.2005.01.002.

[23]

R. D. Mauldin and M. Urbanski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543050.

[24]

I. Palmer, Riemannian Geometry of Compact Metric Spaces, Ph.D. Thesis, Georgia Tech, 2010.

[25]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 1-268.

[26]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.

[27]

J. Pearson and J. Bellissard, Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets, J. Noncommut. Geom., 3 (2009), 447-480. doi: 10.4171/JNCG/43.

[28]

M. Pollicott, A symbolic proof of a theorem of Margulis on geodesic arcs on negatively curved manifolds, Amer. J. Math., 117 (1995), 289-305. doi: 10.2307/2374915.

[29]

M. Pollicott and R. Sharp, Comparison theorems and orbit counting in hyperbolic geometry, Trans. Amer. Math. Soc., 350 (1998), 473-499. doi: 10.1090/S0002-9947-98-01756-5.

[30]

M. Pollicott and R. Sharp, Poincaré series and comparison theorems for variable negative curvature, in Topology, Ergodic Theory, Real Algebraic Geometry, Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, 2001, 229-240.

[31]

D. Ruelle, Thermodynamic Formalism, Second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617546.

[32]

T. Samuel, A Commutative Noncommutative Fractal Geometry, Ph.D. Thesis, St. Andrews University, 2010.

[33]

C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergod. Th. and Dynam. Sys., 6 (1986), 601-625. doi: 10.1017/S0143385700003722.

[34]

R. Sharp, Periodic orbits of hyperbolic flows, in On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004, 73-138.

[35]

R. Sharp, Spectral triples and Gibbs measures for expanding maps on Cantor sets, J. Noncommut. Geom., 6 (2012), 801-817. doi: 10.4171/JNCG/106.

[36]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES, 50 (1979), 171-202.

[37]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379.

[38]

P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math., 152 (1984), 127-140. doi: 10.1007/BF02392194.

[39]

J. Várilly, An Introduction to Noncommutative Geometry, EMS Series of Lectures in Mathematics, European Mathematical Society, Zürich, 2006. doi: 10.4171/024.

[40]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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