American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcds.2020356

Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies

Clark Butler 1, and  Kiho Park 2,

1. 

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
2. 

Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

Received  February 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported by Oswald Veblen fund

We study the norm potentials of Hölder continuous $ \text{GL}_2(\mathbb{R}) $-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant $ \text{GL}_2(\mathbb{R}) $-cocycles as well as fiber-bunched $ \text{GL}_2(\mathbb{R}) $-cocycles. We show that the norm potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose norm potentials have more than one equilibrium states.

Keywords: Subadditive thermodynamic formalism, Norm potentials, Linear cocycles, Equilibrium states, Gibbs property.
Mathematics Subject Classification: 37D35, 37C40, 37H15.
Citation: Clark Butler, Kiho Park. Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020356
References:
[1]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergodic Theory and Dynamical Systems, 16 (1996), 871-927.  doi: 10.1017/S0143385700010117.  Google Scholar

[2]

J. Bochi and E. Garibaldi, Extremal norms for fiber-bunched cocycles, Journal de L'École Polytechnique - Mathématiques, 6 (2019), 947-1004.  doi: 10.5802/jep.109.  Google Scholar

[3]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory and Dynamical Systems, 24 (2004), 1295-1330.  doi: 10.1017/S0143385703000695.  Google Scholar

[4]

R. Bowen, Entropy-expansive maps, Transactions of the American Mathematical Society, 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[5]

R. Bowen, Some systems with unique equilibrium states, Theory of Computing Systems, 8 (1974), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[6]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, 2008.  Google Scholar

[7]

Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete and Continuous Dynamical Systems, 20 (2008), 639-657.  doi: 10.3934/dcds.2008.20.639.  Google Scholar

[8]

Y. Cao, Y. Pesin and Y. Zhao, Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure, Geometric and Functional Analysis, 29 (2019), 1325-1368.  doi: 10.1007/s00039-019-00510-7.  Google Scholar

[9]

B. Call and K. Park, The K-property for subadditive equilibrium states, to appear in Dynamical Systems: An International Journal, arXiv: 2004.13087. Google Scholar

[10]

D. Feng, Equilibrium states for factor maps between subshifts, Advances in Mathematics, 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.  Google Scholar

[11]

D. Feng and A. Käenmäki, Equilibrium states of the pressure function for products of matrices, Discrete and Continuous Dynamical Systems, 30 (2011), 699-708.  doi: 10.3934/dcds.2011.30.699.  Google Scholar

[12]

D. Feng and P. Shmerkin, Non-conformal repellers and the continuity of pressure for matrix cocycles, Geometric and Functional Analysis, 24 (2014), 1101-1128.  doi: 10.1007/s00039-014-0274-7.  Google Scholar

[13]

B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems and criteria of conformality, Journal of Modern Dynamics, 4 (2010), 419-441.  doi: 10.3934/jmd.2010.4.419.  Google Scholar

[14]

B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188.  doi: 10.1007/s10711-012-9808-z.  Google Scholar

[15]

M. Misiurewicz, Topological conditional entropy, Studia Mathematica, 55 (1976), 175-200.  doi: 10.4064/sm-55-2-175-200.  Google Scholar

[16]

K. Park, Quasi-multiplicativity of typical cocycles, Communications in Mathematical Physics, 376 (2020), 1957-2004.  doi: 10.1007/s00220-020-03701-8.  Google Scholar

[17]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Mathematics, 167 (2008), 643-680.  doi: 10.4007/annals.2008.167.643.  Google Scholar

[18]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

References:
[1]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergodic Theory and Dynamical Systems, 16 (1996), 871-927.  doi: 10.1017/S0143385700010117.  Google Scholar

[2]

J. Bochi and E. Garibaldi, Extremal norms for fiber-bunched cocycles, Journal de L'École Polytechnique - Mathématiques, 6 (2019), 947-1004.  doi: 10.5802/jep.109.  Google Scholar

[3]

C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory and Dynamical Systems, 24 (2004), 1295-1330.  doi: 10.1017/S0143385703000695.  Google Scholar

[4]

R. Bowen, Entropy-expansive maps, Transactions of the American Mathematical Society, 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[5]

R. Bowen, Some systems with unique equilibrium states, Theory of Computing Systems, 8 (1974), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[6]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, 2008.  Google Scholar

[7]

Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete and Continuous Dynamical Systems, 20 (2008), 639-657.  doi: 10.3934/dcds.2008.20.639.  Google Scholar

[8]

Y. Cao, Y. Pesin and Y. Zhao, Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure, Geometric and Functional Analysis, 29 (2019), 1325-1368.  doi: 10.1007/s00039-019-00510-7.  Google Scholar

[9]

B. Call and K. Park, The K-property for subadditive equilibrium states, to appear in Dynamical Systems: An International Journal, arXiv: 2004.13087. Google Scholar

[10]

D. Feng, Equilibrium states for factor maps between subshifts, Advances in Mathematics, 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.  Google Scholar

[11]

D. Feng and A. Käenmäki, Equilibrium states of the pressure function for products of matrices, Discrete and Continuous Dynamical Systems, 30 (2011), 699-708.  doi: 10.3934/dcds.2011.30.699.  Google Scholar

[12]

D. Feng and P. Shmerkin, Non-conformal repellers and the continuity of pressure for matrix cocycles, Geometric and Functional Analysis, 24 (2014), 1101-1128.  doi: 10.1007/s00039-014-0274-7.  Google Scholar

[13]

B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems and criteria of conformality, Journal of Modern Dynamics, 4 (2010), 419-441.  doi: 10.3934/jmd.2010.4.419.  Google Scholar

[14]

B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013), 167-188.  doi: 10.1007/s10711-012-9808-z.  Google Scholar

[15]

M. Misiurewicz, Topological conditional entropy, Studia Mathematica, 55 (1976), 175-200.  doi: 10.4064/sm-55-2-175-200.  Google Scholar

[16]

K. Park, Quasi-multiplicativity of typical cocycles, Communications in Mathematical Physics, 376 (2020), 1957-2004.  doi: 10.1007/s00220-020-03701-8.  Google Scholar

[17]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Mathematics, 167 (2008), 643-680.  doi: 10.4007/annals.2008.167.643.  Google Scholar

[18]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[1]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353
[2]

Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300
[3]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[4]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
[5]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[6]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
[7]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[8]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[9]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[10]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[11]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[12]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168
[13]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433
[14]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
[15]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260
[16]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (18)
  • HTML views (65)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences