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 Published  June 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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[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.

[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.
[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.

[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.

[37]

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

[38]

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

[39]

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

[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.

[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.

[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.
[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.

[44]

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

[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.

[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.

[47]

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

[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.

[49]

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

[50]

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

[51]

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

[52]

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

[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.

[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.

[55]

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

[56]

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[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.

[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.
[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.

[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.

[37]

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

[38]

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

[39]

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

[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.

[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.

[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.
[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.

[44]

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

[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.

[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.

[47]

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

[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.

[49]

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

[50]

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

[51]

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

[52]

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

[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.

[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.

[55]

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

[56]

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

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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