# American Institute of Mathematical Sciences

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:
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##### 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. 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. [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. [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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