2020, 16: 155-205. doi: 10.3934/jmd.2020006

Equilibrium measures for some partially hyperbolic systems

1. 

Department of Mathematics, University of Houston, Houston, TX 77204, USA URL: https://www.math.uh.edu/~climenha/

2. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA URL: http://www.math.psu.edu/pesin/

3. 

Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received  July 26, 2019 Revised  March 21, 2020

Fund Project: Partially supported by NSF grant DMS-1554794. Partially supported by NSF grant DMS-1400027. Partially supported by NSF grant DMS-1400027

We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.

Citation: Vaughn Climenhaga, Yakov Pesin, Agnieszka Zelerowicz. Equilibrium measures for some partially hyperbolic systems. Journal of Modern Dynamics, 2020, 16: 155-205. doi: 10.3934/jmd.2020006
References:
[1]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057.  Google Scholar

[2]

A. Arbieto and L. Prudente, Uniqueness of equilibrium states for some partially hyperbolic horseshoes, Discrete Contin. Dyn. Syst., 32 (2012), 27-40.  doi: 10.3934/dcds.2012.32.27.  Google Scholar

[3]

C. Bonatti, S. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 185–244. doi: 10.1007/s10240-009-0021-z.  Google Scholar

[4]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.  Google Scholar

[5]

C. Bonatti and J. Zhang, Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center, arXiv preprint, 2019, arXiv: 1904.05295., Google Scholar

[6]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[7]

R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[8]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, revised ed., with a preface by David Ruelle, edited by Jean-René Chazottes, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008.  Google Scholar

[9]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[10]

K. BurnsD. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys., 108 (2002), 927-942.  doi: 10.1023/A:1019779128351.  Google Scholar

[11]

K. BurnsD. DolgopyatY. Pesin and M. Pollicott, Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn., 2 (2008), 63-81.  doi: 10.3934/jmd.2008.2.63.  Google Scholar

[12]

K. Burns and M. Pollicott, Stable ergodicity and frame flows, Geom. Dedicata, 98 (2003), 189-210.  doi: 10.1023/A:1024057924334.  Google Scholar

[13]

J. BuzziT. FisherM. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems, Ergodic Theory Dynam. Systems, 32 (2012), 63-79.  doi: 10.1017/S0143385710000854.  Google Scholar

[14]

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity, J. Mod. Dyn., 7 (2013), 527-552.  doi: 10.3934/jmd.2013.7.527.  Google Scholar

[15]

V. ClimenhagaT. Fisher and D. J. Thompson, Unique equilibrium states for Bonatti–Viana diffeomorphisms, Nonlinearity, 31 (2018), 2532-2570.  doi: 10.1088/1361-6544/aab1cd.  Google Scholar

[16]

V. ClimenhagaT. Fisher and D. J. Thompson, Equilibrium states for Mañé diffeomorphisms, Ergodic Theory Dynam. Systems, 39 (2019), 2433-2455.  doi: 10.1017/etds.2017.125.  Google Scholar

[17]

V. ClimenhagaY. Pesin and A. Zelerowicz, Equilibrium states in dynamical systems via geometric measure theory, Bull. Amer. Math. Soc. (N.S.), 56 (2019), 569-610.  doi: 10.1090/bull/1659.  Google Scholar

[18]

W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynam. Systems, 25 (2005), 1115-1138.  doi: 10.1017/S0143385704000604.  Google Scholar

[19]

L. J. DíazK. Gelfert and M. Rams, Rich phase transitions in step skew products, Nonlinearity, 24 (2011), 3391-3412.  doi: 10.1088/0951-7715/24/12/005.  Google Scholar

[20]

L. J. DíazK. Gelfert and M. Rams, Abundant rich phase transitions in step-skew products, Nonlinearity, 27 (2014), 2255-2280.  doi: 10.1088/0951-7715/27/9/2255.  Google Scholar

[21]

Lorenzo J. Díaz and To dd Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 29 (2011), 1419-1441.  doi: 10.3934/dcds.2011.29.1419.  Google Scholar

[22]

L. J. DíazT. FisherM. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 32 (2012), 4195-4207.  doi: 10.3934/dcds.2012.32.4195.  Google Scholar

[23]

L. J. Díaz and K. Gelfert, Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions, Fund. Math., 216 (2012), 55-100.  doi: 10.4064/fm216-1-2.  Google Scholar

[24]

E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366.   Google Scholar

[25]

T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math., 160 (2005), 453-499.  doi: 10.1007/s00222-004-0413-0.  Google Scholar

[26]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.  Google Scholar

[27]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissen-schaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[28]

T. Fisher and B. Hasselblatt, Hyperbolic Flows, Zurich Lectures in Advanced Mathematics, 2019. doi: 10.4171/200.  Google Scholar

[29]

U. Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory Dynam. Systems, 9 (1989), 455-464.  doi: 10.1017/S0143385700005095.  Google Scholar

[30]

U. Hamenstädt, Cocycles, Hausdorff measures and cross ratios, Ergodic Theory Dynam. Systems, 17 (1997), 1061-1081.  doi: 10.1017/S0143385797086379.  Google Scholar

[31]

B. Hasselblatt, A new construction of the Margulis measure for Anosov flows, Ergodic Theory Dynam. Systems, 9 (1989), 465-468.  doi: 10.1017/S0143385700005101.  Google Scholar

[32]

N. T. A. Haydn, Canonical product structure of equilibrium states, Random Comput. Dynam., 2 (1994), 79-96.   Google Scholar

[33]

H. HuY. Hua and W. Wu, Unstable entropies and variational principle for partially hyperbolic diffeomorphisms, Adv. Math., 321 (2017), 31-68.  doi: 10.1016/j.aim.2017.09.039.  Google Scholar

[34] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, with a supplementary chapter by A. Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[35]

R. LeplaideurK. Oliveira and I. Rios, Equilibrium states for partially hyperbolic horseshoes, Ergodic Theory Dynam. Systems, 31 (2011), 179-195.  doi: 10.1017/S0143385709000972.  Google Scholar

[36]

R. Leplaideur, Local product structure for equilibrium states, Trans. Amer. Math. Soc.. 352 (2000), 1889–1912. doi: 10.1090/S0002-9947-99-02479-4.  Google Scholar

[37]

G. A. Margulis, Certain measures that are connected with u-flows on compact manifolds, Funkcional. Anal. i Priložen., 4 (1970), 62-76.   Google Scholar

[38]

S. E. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar

[39]

P. Orlik, Seifert Manifolds, Lecture Notes in Mathematics, vol. 291, Springer, 1972.  Google Scholar

[40]

Ya. B. Pesin and Ya. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[41]

Y. Pesin and V. Climenhaga, Open problems in the theory of non-uniform hyperbolicity, Discrete Contin. Dyn. Syst., 27 (2010), 589-607.  doi: 10.3934/dcds.2010.27.589.  Google Scholar

[42] Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar
[43]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[44]

J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754.  doi: 10.2307/2373755.  Google Scholar

[45]

A. Quas and T. Soo, Weak mixing suspension flows over shifts of finite type are universal, J. Mod. Dyn., 6 (2012), 427-449.  doi: 10.3934/jmd.2012.6.427.  Google Scholar

[46]

V. Ramos and J. Siqueira, On equilibrium states for partially hyperbolic horseshoes: Uniqueness and statistical properties, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 347-375.  doi: 10.1007/s00574-017-0027-y.  Google Scholar

[47]

V. Ramos and M. Viana, Equilibrium states for hyperbolic potentials, Nonlinearity, 30 (2017), 825-847.  doi: 10.1088/1361-6544/aa4ec3.  Google Scholar

[48]

F. Rodriguez HertzM. A. Rodriguez HertzA. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves, Ergodic Theory Dynam. Systems, 32 (2012), 825-839.  doi: 10.1017/S0143385711000757.  Google Scholar

[49]

F. Rodriguez HertzM. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Fields Institute Comm., 51 (2007), 35-87.   Google Scholar

[50]

V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55pp.  Google Scholar

[51]

V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk, 22 (1967), 3-56.   Google Scholar

[52]

Ja. G. Sinaĭ, Markov partitions and c-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89.   Google Scholar

[53]

R. Spatzier and D. Visscher, Equilibrium measures for certain isometric extensions of Anosov systems, Ergodic Theory Dynam. Systems, 38 (2018), 1154-1167.  doi: 10.1017/etds.2016.62.  Google Scholar

[54]

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985.  doi: 10.1090/S0002-9939-2011-11040-2.  Google Scholar

[55]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

[56]

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

show all references

References:
[1]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057.  Google Scholar

[2]

A. Arbieto and L. Prudente, Uniqueness of equilibrium states for some partially hyperbolic horseshoes, Discrete Contin. Dyn. Syst., 32 (2012), 27-40.  doi: 10.3934/dcds.2012.32.27.  Google Scholar

[3]

C. Bonatti, S. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 185–244. doi: 10.1007/s10240-009-0021-z.  Google Scholar

[4]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.  Google Scholar

[5]

C. Bonatti and J. Zhang, Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center, arXiv preprint, 2019, arXiv: 1904.05295., Google Scholar

[6]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[7]

R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[8]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, revised ed., with a preface by David Ruelle, edited by Jean-René Chazottes, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008.  Google Scholar

[9]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[10]

K. BurnsD. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys., 108 (2002), 927-942.  doi: 10.1023/A:1019779128351.  Google Scholar

[11]

K. BurnsD. DolgopyatY. Pesin and M. Pollicott, Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn., 2 (2008), 63-81.  doi: 10.3934/jmd.2008.2.63.  Google Scholar

[12]

K. Burns and M. Pollicott, Stable ergodicity and frame flows, Geom. Dedicata, 98 (2003), 189-210.  doi: 10.1023/A:1024057924334.  Google Scholar

[13]

J. BuzziT. FisherM. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems, Ergodic Theory Dynam. Systems, 32 (2012), 63-79.  doi: 10.1017/S0143385710000854.  Google Scholar

[14]

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity, J. Mod. Dyn., 7 (2013), 527-552.  doi: 10.3934/jmd.2013.7.527.  Google Scholar

[15]

V. ClimenhagaT. Fisher and D. J. Thompson, Unique equilibrium states for Bonatti–Viana diffeomorphisms, Nonlinearity, 31 (2018), 2532-2570.  doi: 10.1088/1361-6544/aab1cd.  Google Scholar

[16]

V. ClimenhagaT. Fisher and D. J. Thompson, Equilibrium states for Mañé diffeomorphisms, Ergodic Theory Dynam. Systems, 39 (2019), 2433-2455.  doi: 10.1017/etds.2017.125.  Google Scholar

[17]

V. ClimenhagaY. Pesin and A. Zelerowicz, Equilibrium states in dynamical systems via geometric measure theory, Bull. Amer. Math. Soc. (N.S.), 56 (2019), 569-610.  doi: 10.1090/bull/1659.  Google Scholar

[18]

W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynam. Systems, 25 (2005), 1115-1138.  doi: 10.1017/S0143385704000604.  Google Scholar

[19]

L. J. DíazK. Gelfert and M. Rams, Rich phase transitions in step skew products, Nonlinearity, 24 (2011), 3391-3412.  doi: 10.1088/0951-7715/24/12/005.  Google Scholar

[20]

L. J. DíazK. Gelfert and M. Rams, Abundant rich phase transitions in step-skew products, Nonlinearity, 27 (2014), 2255-2280.  doi: 10.1088/0951-7715/27/9/2255.  Google Scholar

[21]

Lorenzo J. Díaz and To dd Fisher, Symbolic extensions and partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 29 (2011), 1419-1441.  doi: 10.3934/dcds.2011.29.1419.  Google Scholar

[22]

L. J. DíazT. FisherM. J. Pacifico and J. L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 32 (2012), 4195-4207.  doi: 10.3934/dcds.2012.32.4195.  Google Scholar

[23]

L. J. Díaz and K. Gelfert, Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions, Fund. Math., 216 (2012), 55-100.  doi: 10.4064/fm216-1-2.  Google Scholar

[24]

E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366.   Google Scholar

[25]

T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math., 160 (2005), 453-499.  doi: 10.1007/s00222-004-0413-0.  Google Scholar

[26]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.  Google Scholar

[27]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissen-schaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[28]

T. Fisher and B. Hasselblatt, Hyperbolic Flows, Zurich Lectures in Advanced Mathematics, 2019. doi: 10.4171/200.  Google Scholar

[29]

U. Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory Dynam. Systems, 9 (1989), 455-464.  doi: 10.1017/S0143385700005095.  Google Scholar

[30]

U. Hamenstädt, Cocycles, Hausdorff measures and cross ratios, Ergodic Theory Dynam. Systems, 17 (1997), 1061-1081.  doi: 10.1017/S0143385797086379.  Google Scholar

[31]

B. Hasselblatt, A new construction of the Margulis measure for Anosov flows, Ergodic Theory Dynam. Systems, 9 (1989), 465-468.  doi: 10.1017/S0143385700005101.  Google Scholar

[32]

N. T. A. Haydn, Canonical product structure of equilibrium states, Random Comput. Dynam., 2 (1994), 79-96.   Google Scholar

[33]

H. HuY. Hua and W. Wu, Unstable entropies and variational principle for partially hyperbolic diffeomorphisms, Adv. Math., 321 (2017), 31-68.  doi: 10.1016/j.aim.2017.09.039.  Google Scholar

[34] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, with a supplementary chapter by A. Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[35]

R. LeplaideurK. Oliveira and I. Rios, Equilibrium states for partially hyperbolic horseshoes, Ergodic Theory Dynam. Systems, 31 (2011), 179-195.  doi: 10.1017/S0143385709000972.  Google Scholar

[36]

R. Leplaideur, Local product structure for equilibrium states, Trans. Amer. Math. Soc.. 352 (2000), 1889–1912. doi: 10.1090/S0002-9947-99-02479-4.  Google Scholar

[37]

G. A. Margulis, Certain measures that are connected with u-flows on compact manifolds, Funkcional. Anal. i Priložen., 4 (1970), 62-76.   Google Scholar

[38]

S. E. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar

[39]

P. Orlik, Seifert Manifolds, Lecture Notes in Mathematics, vol. 291, Springer, 1972.  Google Scholar

[40]

Ya. B. Pesin and Ya. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[41]

Y. Pesin and V. Climenhaga, Open problems in the theory of non-uniform hyperbolicity, Discrete Contin. Dyn. Syst., 27 (2010), 589-607.  doi: 10.3934/dcds.2010.27.589.  Google Scholar

[42] Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar
[43]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[44]

J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754.  doi: 10.2307/2373755.  Google Scholar

[45]

A. Quas and T. Soo, Weak mixing suspension flows over shifts of finite type are universal, J. Mod. Dyn., 6 (2012), 427-449.  doi: 10.3934/jmd.2012.6.427.  Google Scholar

[46]

V. Ramos and J. Siqueira, On equilibrium states for partially hyperbolic horseshoes: Uniqueness and statistical properties, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 347-375.  doi: 10.1007/s00574-017-0027-y.  Google Scholar

[47]

V. Ramos and M. Viana, Equilibrium states for hyperbolic potentials, Nonlinearity, 30 (2017), 825-847.  doi: 10.1088/1361-6544/aa4ec3.  Google Scholar

[48]

F. Rodriguez HertzM. A. Rodriguez HertzA. Tahzibi and R. Ures, Maximizing measures for partially hyperbolic systems with compact center leaves, Ergodic Theory Dynam. Systems, 32 (2012), 825-839.  doi: 10.1017/S0143385711000757.  Google Scholar

[49]

F. Rodriguez HertzM. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Fields Institute Comm., 51 (2007), 35-87.   Google Scholar

[50]

V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55pp.  Google Scholar

[51]

V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk, 22 (1967), 3-56.   Google Scholar

[52]

Ja. G. Sinaĭ, Markov partitions and c-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89.   Google Scholar

[53]

R. Spatzier and D. Visscher, Equilibrium measures for certain isometric extensions of Anosov systems, Ergodic Theory Dynam. Systems, 38 (2018), 1154-1167.  doi: 10.1017/etds.2016.62.  Google Scholar

[54]

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985.  doi: 10.1090/S0002-9939-2011-11040-2.  Google Scholar

[55]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

[56]

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

Figure 6.1.  Proving Lemma 6.2
Figure 8.1.  Birkhoff averages are essentially constant
Figure 8.2.  The partitions $ {\mathcal{R}} $ and $ {\mathcal{R}}_n $, and the choice of $ \epsilon_R $
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