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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Thermodynamic formalism for countable to one Markov systems


Author: Michiko Yuri
Journal: Trans. Amer. Math. Soc. 355 (2003), 2949-2971
MSC (2000): Primary 28D99, 28D20, 58C40, 58E30, 37A40, 37A30, 37C30, 37D35, 37F10, 37A45
DOI: https://doi.org/10.1090/S0002-9947-03-03269-0
Published electronically: March 19, 2003
MathSciNet review: 1975407
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Abstract: For countable to one transitive Markov systems we establish thermodynamic formalism for non-Hölder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.


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  • 1. J. Aaronson, M. Denker and M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548. MR 94g:58116
  • 2. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Mathematics 470, Springer, 1975. MR 56:1364
  • 3. M. Denker, F. Przytycki and M. Urbanski, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory and Dynamical Systems 16 (1996), 255-266. MR 97e:58197
  • 4. M. Denker and M. Urbanski, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), 563-587. MR 92k:58155
  • 5. M. Denker and M. Yuri, A note on the construction of nonsingular Gibbs measures, Colloquium Mathematicum 84/85 (2000), 377-383. MR 2001k:37013
  • 6. M. Denker and M. Yuri, Partially defined infinite iterated functional systems, In preparation.
  • 7. D. Fiebig, U. Fiebig and M. Yuri, Pressures and Equilibrium states for countable Markov shifts, To appear in Israel J. Math.
  • 8. P. H. Hanus, R. D. Mauldin and M. Urbanski, Thermodynamic formalism and multi-fractal analysis of conformal infinite iterated functional systems, Acta Math. Hungar. 96 (2002), 27-98.
  • 9. R. D. Mauldin and M. Urbanski, Parabolic iterated function systems, Ergodic Theory and Dynamical Systems 20 (2000), 1423-1447. MR 2001m:37047
  • 10. Omri Sarig, Thermodynamic formalism for countable Markov shifts., Ergodic Theory and Dynamical Systems 19 (1999), 1565-1593. MR 2000m:37009
  • 11. Y. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago University Press, 1997. MR 99b:58003
  • 12. M. Pollicott and M. Yuri, Zeta functions for certain multi-dimensional nonhyperbolic maps, Nonlinearity 14 (2001), 1265-1278. MR 2002h:37036
  • 13. F. Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford University Press, Oxford, 1995. MR 97h:11083
  • 14. P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121-153. MR 57:6371
  • 15. M. Yuri, On a Bernoulli property for multi-dimensional mappings with finite range structure, Tokyo J. Math 9 (1986), 457-485. MR 88d:28023
  • 16. M. Yuri, On the convergence to equilibrium states for certain nonhyperbolic systems, Ergodic Theory and Dynamical Systems 17 (1997), 977-1000. MR 98f:58155
  • 17. M. Yuri, Zeta functions for certain nonhyperbolic systems and topological Markov approximations, Ergodic Theory and Dynamical Systems 18 (1998), 1589-1612. MR 2000j:37024
  • 18. M. Yuri, Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory and Dynamical Systems 19 (1999), 1365-1378. MR 2001a:37012
  • 19. M. Yuri, Statistical properties for nonhyperbolic maps with finite range structure, Trans. Amer. Math. Soc. 352 (2000), 2369-2388. MR 2000j:37009
  • 20. M. Yuri, Weak Gibbs measures for certain nonhyperbolic systems, Ergodic Theory and Dynamical Systems 20 (2000), 1495-1518. MR 2002d:37011
  • 21. M. Yuri, On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points., Nonlinearity 15 (2002), 429-445. MR 2002k:37006
  • 22. M. Yuri, Multifractal Analysis of weak Gibbs measures for Intermittent Systems., Commun. Math. Phys. 230 (2002), 365-388.

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Additional Information

Michiko Yuri
Affiliation: Department of Business Administration, Sapporo University, Nishioka, Toyohira-ku, Sapporo 062-8520, Japan
Email: yuri@math.sci.hokudai.ac.jp, yuri@mail-ext.sapporo-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-03-03269-0
Received by editor(s): March 28, 2002
Received by editor(s) in revised form: September 10, 2002
Published electronically: March 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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