Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Equilibriums of some non-Hölder potentials

Author(s): Huyi Hu
Journal: Trans. Amer. Math. Soc. 360 (2008), 2153-2190.
MSC (2000): Primary 37C40, 37A60; Secondary 28D05
Posted: October 22, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We consider one-sided subshifts $ \sigma $ with some potential functions $ \varphi $ which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures $ \nu $ and invariant measures $ \mu $ absolutely continuous with respect to $ \nu $, where $ \mu $ may be finite or infinite. We show that the systems $ (\sigma , \mu )$ are exact, and $ \mu $ are weak Gibbs measures and equilibriums for $ \varphi $. We also discuss uniqueness of equilibriums and phase transition.

These results can be applied to some expanding dynamical systems with an indifferent fixed point.

References:

[A]
J. Aaronson, An introduction to infinite ergodic theory (Mathematical Surveys and Monographs, 50), American Mathematical Society, Providence, RI, 1997. MR 1450400 (99d:28025)

[ADU]
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 1107025 (94g:58116)

[B]
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, New York, 1975. MR 0442989 (56:1364)

[DU1]
M. Denker and M. Urbanski, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134. MR 1092887 (92a:58112)

[DU2]
M. Denker and M. Urbanski, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. (2) 43 (1991), 107-118. MR 1099090 (92k:58153)

[DU3]
M. Denker and M. Urbanski, Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), 561-579. MR 1129999 (92k:58154)

[DU4]
M. Denker and M. Urbanski, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53-66. MR 1162398 (93d:58133)

[DS]
N. Dunford and J. Schwartz, Linear Operator Part I, Interscience Publishers, New York, 1958. MR 0117523 (22:8302)

[FL]
A. Fisher and A. Lopes, Polynomial decay of correlations and the central limit theorem for the equilibrium states of a non-Hölder potential, Nonlinearity 14 (2001), 1071-1104. MR 1862813 (2002g:37032)

[G]
S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math. 139 (2004), 29-65. MR 2041223 (2004m:37011)

[Ha]
N. Haydn, Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems 19 (1999), 657-669. MR 1695914 (2000f:37055)

[HH]
N. Haydn and H. Hu, Convergence of the transfer operator for sigma finite measures, Preprint.

[Ho]
F. Hofbauer, Examples for the nonuniqueness of the equilibrium state, Trans. Amer. Math. Soc. 228 (1977), 223-241. MR 0435352 (55:8312)

[H1]
H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems 24 (2004), 495-524. MR 2054191 (2005a:37064)

[H2]
H. Hu, Equilibrium states with polynomial decay of correlations, Preprint.

[HV]
H. Hu and S. Vaienti, Absolutely continuous invariant measures for nonuniformly expanding maps, Preprint.

[KMS]
A. Kondah, V. Maume, and B. Schmitt, Vitesse de convergence vers l'état d'équilibre pour des dynamiques markoviennes non höldriennes (French) [Rate of convergence to equilibrium for non-Holder Markov dynamics], Ann. Inst. H. Poincare Probab. Statist. 33 (1997), 675-695. MR 1484537 (99e:28032)

[Ke]
G. Keller, Equilibrium States in ergodic Theory, Cambridge Univ. Press, Cambridge, 1998. MR 1618769 (99e:28022)

[Kr]
U. Krengel, Entropy of conservative transformations, Z. Wahrs. Verw. Geb. 7 (1967), 161-181. MR 0218522 (36:1608)

[Li]
M. Lin, Mixing of Cartesian squares of positive operators, Israel J. Math. 11 (1972), 349-354. MR 0308365 (46:7479)

[LiSV]
C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), 671-685. MR 1695915 (2000d:37029)

[M]
V. Maume-Deschamps, Equilibrium states for non-Hölderian random dynamical systems, Random Comput. Dynam. 5 (1997), 319-335. MR 1483873 (99a:58104)

[N]
T. Nakata, Pianigiani-Yorke measures for non-Hölder continuous potentials, Hiroshima Math. J. 28 (1998), 95-111. MR 1612822 (99c:58095)

[Pi]
G. Pianigiani, First return map and invariant measures, Israel J. Math. 35 (1980), 32-48. MR 576460 (82a:58031)

[PY]
M. Pollicott and M. Yuri, Statistical Properties of maps with indifferent periodic points, Comm. Math. Phys. 17 (2001), 503-520. MR 1822105 (2002h:37049)

[Pr]
F. Przytycki, On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions, Boll. Bras. Soc. Mat. 20 (1990), 95-125. MR 1143178 (93b:58120)

[R]
V. Rohlin, Lectures on the entropy theory of measure-preserving transformations, Russ. Math. Surv. 22 (1967), 1-54. MR 0217258 (36:349)

[S1]
O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), 1565-1593. MR 1738951 (2000m:37009)

[S2]
O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), 555-577. MR 1822107 (2002b:37040)

[S3]
O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), 1751-1758. MR 1955261 (2004b:37056)

[S4]
O. Sarig, Subexponential decay of correlations, Invent. Math. 150 (2002), 629-653. MR 1946554 (2004e:37010)

[T]
M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent periodic points, Israel J. Math. 37 (1980), 303-314. MR 599464 (82f:28021)

[U1]
M. Urbanski, Parabolic Cantor sets, Fundamenta Mathematicae 151 (1996), 241-277. MR 1424576 (98f:58076)

[U2]
M. Urbanski, Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 281-321. MR 1978566 (2004f:37063)

[W1]
P. Walters, Invariant measure and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121-153. MR 0466493 (57:6371)

[W2]
P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1981. MR 648108 (84e:28017)

[Y1]
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), 585-650. MR 1637655 (99h:58140)

[Y2]
L.-S. Young, Recurrence time and rate of mixing, Israel J. Math. 110 (1999), 153-188. MR 1750438 (2001j:37062)

[Yu1]
M. Yuri, Thermodynamic formalism for certain nonhyperbolic maps, Ergod. Th. & Dynam. Sys. 19 (1999), 1365-1378. MR 1721626 (2001a:37012)

[Yu2]
M. Yuri, Weak Gibbs measure for certain nonhyperbolic systems, Ergodic Theory Dynam. Systems 20 (2000), 1495-1518. MR 1786726 (2002d:37011)

[Yu3]
M. Yuri, Weak Gibbs measures for intermittent systems and weakly Gibbsian states in statistical mechanics, Comm. Math. Phys. 241 (2003), 453-466. MR 2013806 (2005a:37045)

[Yu4]
M. Yuri, Phase transition, non-gibbsianness and subexponential instability, Preprint. MR 2158408 (2006f:37031)

[Z]
R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems 20 (2000), 1519-1549. MR 1786727 (2001h:37014)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37C40, 37A60, 28D05

Retrieve articles in all Journals with MSC (2000): 37C40, 37A60, 28D05

Additional Information:

Huyi Hu
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: hu@math.msu.edu

DOI: 10.1090/S0002-9947-07-04412-1
PII: S 0002-9947(07)04412-1
Keywords: Potential, equilibrium, invariant measure, exactness, ergodicity, weak Gibbs measure
Received by editor(s): January 9, 2006
Received by editor(s) in revised form: June 10, 2006
Posted: October 22, 2007
Additional Notes: Part of this work was done when the author was at Penn State University and the University of Southern California. This work was supported by NSF under grants DMS-9970646 and DMS-0240097.
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google