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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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