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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 |
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.
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. 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. |
[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. 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. 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. |
[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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