April  2017, 10(2): 335-352. doi: 10.3934/dcdss.2017016

A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory

1. 

FB03-Mathematik und Informatik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany

2. 

Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany

Received  December 2015 Revised  November 2016 Published  January 2017

We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.

Citation: Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016
References:
[1]

G. Alsmeyer, Erneuerungstheorie: Analyse Stochastischer Regenerationsschemata Teubner Skripten zur mathematischen Stochastik, Teubner B. G. GmbH, 1991. doi: 10.1007/978-3-663-09977-2.

[2]

S. Asmussen, Applied Probability and Queues. 2nd Revised and Extended ed. 2nd edition, New York, NY: Springer, 2003, URL http://dx.doi.org/10.1007/b97236.

[3]

V. I. Bogachev, Measure Theory. Vol. Ⅰ Ⅱ, Springer-Verlag, Berlin, 2007, URL http://dx.doi.org/10.1007/978-3-540-34514-5. doi: 10.1007/978-3-540-34514-5.

[4]

K. Falconer, Techniques in Fractal Geometry John Wiley & Sons, Ltd. , Chichester, 1997.

[5]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. Ⅱ. Second edition, John Wiley & Sons Inc. , New York, 1971.

[6]

P. Hanus and M. Urba«ski, A new class of positive recurrent functions, in Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), vol. 246 of Contemp. Math. , Amer. Math. Soc. , Providence, RI, 1999,123-135, URL http://dx.doi.org/10.1090/conm/246/03779. doi: 10.1090/conm/246/03779.

[7]

J. Jaerisch, M. Kesseböhmer and S. Lamei, Induced topological pressure for countable state Markov shifts Stoch. Dyn. 14 (2014), 1350016, 31pp, URL http://dx.doi.org/10.1142/S0219493713500160. doi: 10.1142/S0219493713500160.

[8]

T. Kato, Perturbation Theory for Linear Operators. Reprint of the corr. print. of the 2nd ed. 1980 edition, Berlin: Springer-Verlag, 1995.

[9]

M. Kesseböhmer and S. Kombrink, Fractal curvature measures and Minkowski content for self-conformal subsets of the real line, Adv. Math. , 230 (2012), 2474-2512, URL http://dx.doi.org/10.1016/j.aim.2012.04.023. doi: 10.1016/j.aim.2012.04.023.

[10]

M. Kesseböhmer and S. Kombrink, Minkowski content and fractal Euler characteristic for conformal graph directed systems, J. Fractal Geom. , 2 (2015), 171-227, URL http://dx.doi.org/10.4171/JFG/19. doi: 10.4171/JFG/19.

[11]

M. Kesseböhmer and S. Kombrink, Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings, preprint, (2017), 1-28.

[12]

S. Kombrink, Fractal curvature measures and Minkowski content for limit sets of conformal function systems PhD Thesis, Universität Bremen, 2011, URL http://nbn-resolving.de/urn:nbn:de:gbv:46-00102477-19.

[13]

S. Kombrink, Renewal theorems for a class of processes with dependent interarrival times, preprint, arXiv: 1512. 08351, 1-35.

[14]

S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. , 37 (1988), 699-710, URL http://dx.doi.org/10.1512/iumj.1988.37.37034. doi: 10.1512/iumj.1988.37.37034.

[15]

S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. , 163 (1989), 1-55, URL http://dx.doi.org/10.1007/BF02392732. doi: 10.1007/BF02392732.

[16]

R. D. Mauldin and M. Urbański, Graph Directed Markov Systems vol. 148 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2003, URL http://dx.doi.org/10.1017/CBO9780511543050, Geometry and dynamics of limit sets. doi: 10.1017/CBO9780511543050.

[17]

W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, no. 187-188 in Astérisque, Société Mathématique de France, Paris, 1990.

[18]

M. Pollicott, A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergodic Theory Dyn. Syst. , 4 (1984), 135-146, URL http://dx.doi.org/10.1017/S0143385700002327/ doi: 10.1017/S0143385700002327.

[19]

O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. , 131 (2003), 1751-1758, URL http://dx.doi.org/10.1090/S0002-9939-03-06927-2. doi: 10.1090/S0002-9939-03-06927-2.

show all references

References:
[1]

G. Alsmeyer, Erneuerungstheorie: Analyse Stochastischer Regenerationsschemata Teubner Skripten zur mathematischen Stochastik, Teubner B. G. GmbH, 1991. doi: 10.1007/978-3-663-09977-2.

[2]

S. Asmussen, Applied Probability and Queues. 2nd Revised and Extended ed. 2nd edition, New York, NY: Springer, 2003, URL http://dx.doi.org/10.1007/b97236.

[3]

V. I. Bogachev, Measure Theory. Vol. Ⅰ Ⅱ, Springer-Verlag, Berlin, 2007, URL http://dx.doi.org/10.1007/978-3-540-34514-5. doi: 10.1007/978-3-540-34514-5.

[4]

K. Falconer, Techniques in Fractal Geometry John Wiley & Sons, Ltd. , Chichester, 1997.

[5]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. Ⅱ. Second edition, John Wiley & Sons Inc. , New York, 1971.

[6]

P. Hanus and M. Urba«ski, A new class of positive recurrent functions, in Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), vol. 246 of Contemp. Math. , Amer. Math. Soc. , Providence, RI, 1999,123-135, URL http://dx.doi.org/10.1090/conm/246/03779. doi: 10.1090/conm/246/03779.

[7]

J. Jaerisch, M. Kesseböhmer and S. Lamei, Induced topological pressure for countable state Markov shifts Stoch. Dyn. 14 (2014), 1350016, 31pp, URL http://dx.doi.org/10.1142/S0219493713500160. doi: 10.1142/S0219493713500160.

[8]

T. Kato, Perturbation Theory for Linear Operators. Reprint of the corr. print. of the 2nd ed. 1980 edition, Berlin: Springer-Verlag, 1995.

[9]

M. Kesseböhmer and S. Kombrink, Fractal curvature measures and Minkowski content for self-conformal subsets of the real line, Adv. Math. , 230 (2012), 2474-2512, URL http://dx.doi.org/10.1016/j.aim.2012.04.023. doi: 10.1016/j.aim.2012.04.023.

[10]

M. Kesseböhmer and S. Kombrink, Minkowski content and fractal Euler characteristic for conformal graph directed systems, J. Fractal Geom. , 2 (2015), 171-227, URL http://dx.doi.org/10.4171/JFG/19. doi: 10.4171/JFG/19.

[11]

M. Kesseböhmer and S. Kombrink, Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings, preprint, (2017), 1-28.

[12]

S. Kombrink, Fractal curvature measures and Minkowski content for limit sets of conformal function systems PhD Thesis, Universität Bremen, 2011, URL http://nbn-resolving.de/urn:nbn:de:gbv:46-00102477-19.

[13]

S. Kombrink, Renewal theorems for a class of processes with dependent interarrival times, preprint, arXiv: 1512. 08351, 1-35.

[14]

S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. , 37 (1988), 699-710, URL http://dx.doi.org/10.1512/iumj.1988.37.37034. doi: 10.1512/iumj.1988.37.37034.

[15]

S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. , 163 (1989), 1-55, URL http://dx.doi.org/10.1007/BF02392732. doi: 10.1007/BF02392732.

[16]

R. D. Mauldin and M. Urbański, Graph Directed Markov Systems vol. 148 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2003, URL http://dx.doi.org/10.1017/CBO9780511543050, Geometry and dynamics of limit sets. doi: 10.1017/CBO9780511543050.

[17]

W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, no. 187-188 in Astérisque, Société Mathématique de France, Paris, 1990.

[18]

M. Pollicott, A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergodic Theory Dyn. Syst. , 4 (1984), 135-146, URL http://dx.doi.org/10.1017/S0143385700002327/ doi: 10.1017/S0143385700002327.

[19]

O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. , 131 (2003), 1751-1758, URL http://dx.doi.org/10.1090/S0002-9939-03-06927-2. doi: 10.1090/S0002-9939-03-06927-2.

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