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May  2021, 41(5): 2141-2166. 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  May 2021 Early access  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, 2021, 41 (5) : 2141-2166. 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

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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

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