May  2021, 41(5): 2141-2166. doi: 10.3934/dcds.2020356

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

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.

Citation: Clark Butler, Kiho Park. Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies. Discrete and 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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[7]

Y. CaoD. 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.

[8]

Y. CaoY. 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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[18]

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

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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[7]

Y. CaoD. 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.

[8]

Y. CaoY. 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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[18]

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

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