Thermodynamic formalism for countable to one Markov systems
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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.References
- Jon Aaronson, Manfred Denker, and Mariusz Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 495–548. MR 1107025, DOI 10.1090/S0002-9947-1993-1107025-2
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
- Manfred Denker, Feliks Przytycki, and Mariusz Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 255–266. MR 1389624, DOI 10.1017/S0143385700008804
- Manfred Denker and Mariusz Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), no. 2, 563–587. MR 1014246, DOI 10.1090/S0002-9947-1991-1014246-4
- Manfred Denker and Michiko Yuri, A note on the construction of nonsingular Gibbs measures. part 2, Colloq. Math. 84/85 (2000), no. part 2, 377–383. Dedicated to the memory of Anzelm Iwanik. MR 1784215, DOI 10.4064/cm-84/85-2-377-383
- M. Denker and M. Yuri, Partially defined infinite iterated functional systems, In preparation.
- D. Fiebig, U. Fiebig and M. Yuri, Pressures and Equilibrium states for countable Markov shifts, To appear in Israel J. Math.
- P. H. Hanus, R. D. Mauldin and M. Urbański, Thermodynamic formalism and multi-fractal analysis of conformal infinite iterated functional systems, Acta Math. Hungar. 96 (2002), 27-98.
- R. D. Mauldin and M. Urbański, Parabolic iterated function systems, Ergodic Theory Dynam. Systems 20 (2000), no. 5, 1423–1447. MR 1786722, DOI 10.1017/S0143385700000778
- Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593. MR 1738951, DOI 10.1017/S0143385799146820
- Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
- Mark Pollicott and Michiko Yuri, Zeta functions for certain multi-dimensional non-hyperbolic maps, Nonlinearity 14 (2001), no. 5, 1265–1278. MR 1862820, DOI 10.1088/0951-7715/14/5/317
- Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1419320
- Peter Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121–153. MR 466493, DOI 10.1090/S0002-9947-1978-0466493-1
- Michiko Yuri, On a Bernoulli property for multidimensional mappings with finite range structure, Tokyo J. Math. 9 (1986), no. 2, 457–485. MR 875201, DOI 10.3836/tjm/1270150732
- Michiko Yuri, On the convergence to equilibrium states for certain non-hyperbolic systems, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 977–1000. MR 1468111, DOI 10.1017/S0143385797086240
- Michiko Yuri, Zeta functions for certain non-hyperbolic systems and topological Markov approximations, Ergodic Theory Dynam. Systems 18 (1998), no. 6, 1589–1612. MR 1658607, DOI 10.1017/S0143385798117972
- Michiko Yuri, Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory Dynam. Systems 19 (1999), no. 5, 1365–1378. MR 1721626, DOI 10.1017/S0143385799141695
- Michiko Yuri, Statistical properties for nonhyperbolic maps with finite range structure, Trans. Amer. Math. Soc. 352 (2000), no. 5, 2369–2388. MR 1695039, DOI 10.1090/S0002-9947-00-02579-4
- Michiko Yuri, Weak Gibbs measures for certain non-hyperbolic systems, Ergodic Theory Dynam. Systems 20 (2000), no. 5, 1495–1518. MR 1786726, DOI 10.1017/S014338570000081X
- Michiko Yuri, On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points, Nonlinearity 15 (2002), no. 2, 429–445. MR 1888859, DOI 10.1088/0951-7715/15/2/311
- M. Yuri, Multifractal Analysis of weak Gibbs measures for Intermittent Systems., Commun. Math. Phys. 230 (2002), 365-388.
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
- Received by editor(s): March 28, 2002
- Received by editor(s) in revised form: September 10, 2002
- Published electronically: March 19, 2003
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1975407