Abstract
We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.
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References
Ahlfors L V. Lectures on Quasiconformal Mappings. Mathematical Studies, vol. 10. Toronto-New York-London: D. Van Nostrand, 1966
Bowen R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Berlin: Springer-Verlag, 1975
Cui G, Gardiner F, Jiang Y. Scaling functions for degree 2 circle endomorphisms. Contemp Math, 2004, 355: 147–163
Cui G, Jiang Y, Quas A. Scaling functions, Gibbs measures, and Teichmüller spaces of circle endomorphisms. Discrete Contin Dyn Syst, 1999, 3: 535–552
Gardiner F. Approximation of infinite dimensional Teichmüller spaces. Trans Amer Math Soc, 1984, 282: 367–383
Gardiner F. Teichmüller Theory and Quadratic Differentials. New York: John Wiley & Sons, 1987
Gardiner F, Jiang Y. Asymptotically affine and asymptotically conformal circle endomorphisms. RIMS Kôkyûroku Bessatsu, 2010, 17: 37–53
Gardiner F, Sullivan D. Symmetric structures on a closed curve. Amer J Math, 1992, 114: 683–736
Hu H, Jiang M, Jiang Y. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete Contin Dyn Syst, 2008, 22: 215–234
Hu H, Jiang M, Jiang Y. Infimum of the metric entropy of volume preserving Anosov systems. Discrete Contin Dyn Syst, 2017, 37: 4767–4783
Hu J, Jiang Y, Wang Z. Kobayashi’s and Teichmüller’s metrics on the Teichmüller space of symmetric circle homeomorphisms. Acta Math Sin (Engl Ser), 2011, 27: 617–624
Jiang Y. Renormalization and Geometry in One-Dimensional and Complex Dynamics. Advanced Series in Nonlinear Dynamics, vol. 10. River Edge: World Scientific, 1996
Jiang Y. A proof of the existence and simplicity of a maximal eigenvalue for Ruelle-Perron-Frobenius operators. Lett Math Phys, 1999, 48: 211–219
Jiang Y. Nanjing Lecture Notes in Dynamical Systems. Part One: Transfer Operators in Thermodynamical Formalism. http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/JiangNJLectureIFM.pdf, 2000
Jiang Y. Metric invariants in dynamical systems. J Dynam Differential Equations, 2005, 17: 51–71
Jiang Y. On a question of Katok in one-dimensional case. Discrete Contin Dyn Syst, 2009, 24: 1209–1213
Jiang Y. Differential rigidity and applications in one-dimensional dynamics. In: Dynamics, Games and Science I. Springer Proceedings in Mathematics, vol. 1. Berlin-Heidelberg: Springer, 2011, 487–502
Jiang Y. Symmetric invariant measures. Contemp Math, 2012, 575: 211–218
Jiang Y. Kobayashi’s and Teichmüller’s metrics and Bers complex manifold structure on circle diffeomorphisms. Acta Math Sin (Engl Ser), 2020, 36: 245–272
Jiang Y, Ruelle D. Analyticity of the susceptibility function for unimodal Markovian maps of the interval. Nonlinearity, 2005, 18: 2447–2453
Keane M. Strongly mixing g-measures. Invent Math, 1972, 16: 309–324
Ledrappier F. Principe variationel et systemes dynamiques symboliques. Z Wahrscheinlichkeitstheorie Verw Gebiete, 1974, 30: 185–202
Reimann M. Ordinary differential equations and quasiconformal mappings. Invent Math, 1976, 33: 247–270
Royden H. Automorphisms and isometries of Teichmuüller space. In: Advances in the Theory of Riemann Surfaces. Stony Brook Conference. Annals of Mathematics Studies, vol. 66. Princeton: Princeton University Press, 1971, 369–383
Ruelle D. Statistical mechanics of a one-dimensional lattice gas. Comm Math Phys, 1968, 9: 267–278
Ruelle D. A measure associated with Axiom A attractors. Amer J Math, 1976, 98: 619–654
Ruelle D. Differentiating the absolutely continuous invariant measure of an interval map f with respect to f. Comm Math Phys, 2005, 258: 445–453
Sinai Y G. Markov partitions and C-diffeomorphisms. Funct Anal Appl, 1968, 2: 61–82
Sinai Y G. Gibbs measures in ergodic theory. Russian Math Surveys, 1972, 27: 21–69
Walters P. Ruelle’s operator theorem and (g-measures. Trans Amer Math Soc, 1975, 214: 375–387
Acknowledgements
This work was supported by National Science Foundation of USA (Grant No. DMS-1747905), the Simons Foundation (Grant No. 523341), Professional Staff Congress of the City University of New York Enhanced Award (Grant No. 62777-00 50) and National Natural Science Foundation of China (Grant No. 11571122). The author thanks his student John Adamski and colleague Sudeb Mitra for proofreading the abstract and the introduction of this paper.
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In Memory of Professor Shantao Liao
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Jiang, Y. Geometric Gibbs theory. Sci. China Math. 63, 1777–1824 (2020). https://doi.org/10.1007/s11425-019-1638-6
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DOI: https://doi.org/10.1007/s11425-019-1638-6
Keywords
- geometric Gibbs measure
- continuous potential
- smooth potential
- Teichmüller’s metric
- maximum metric
- Kobayashi’s metric
- symmetric rigidity
- complex Banach manifold