J. Alves, C. Bonatti, and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Inventiones mathematicae, vol.140, pp.351-398, 2000.

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov, vol.90, 1967.

A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, vol.27, pp.1399-1417, 2007.

V. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, vol.18, pp.13-40, 1963.

A. Avila, J. Bochi, and A. Wilkinson, Nonuniform center bunching and the genericity of ergodicity among C 1 partially hyperbolic symplectomorphisms, Ann. Sci. Éc. Norm. Supér, vol.42, pp.931-979, 2009.

A. Avila, S. Crovisier, and A. Wilkinson, C 1 density of stable ergodicity, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02367511

A. Avila, S. Crovisier, and A. Wilkinson, Diffeomorphisms with positive metric entropy, Publ. Math. IHES, vol.124, pp.319-347, 2016.

A. Avila, J. Santamaria, and M. Viana, Holonomy invariance: rough regularity and applications to Lyapunov exponents, Astérisque, vol.358, pp.13-74, 2013.

A. Avila and M. Viana, Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math, vol.181, pp.115-189, 2010.

P. G. Barrientos and D. Malicet, Extremal exponents of random products of conservative diffeomorphisms, 2018.

S. and B. Ovadia, Generalized SRB measures, physical properties, and thermodynamic formalism of smooth hyperbolic systems, 2019.

P. Berger and P. Carrasco, Non-uniformly hyperbolic diffeomorphisms derived from the standard map, Comm. Math. Phys, vol.329, pp.239-262, 2014.

P. Berger and D. Turaev, On Herman's positive entropy conjecture, 2017.

G. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci, vol.17, pp.656-660, 1931.

A. Blumenthal, J. Xue, and L. S. Young, Lyapunov exponents for random perturbations of some area-preserving maps including the standard map, Ann. of Math, vol.185, pp.285-310, 2017.

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, vol.22, pp.1667-1696, 2002.

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math, vol.143, pp.357-396, 1996.

C. Bonatti, L. Díaz, and E. Pujals, A C 1 -generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math, vol.158, pp.355-418, 2003.

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math, vol.115, pp.157-193, 2000.

R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math, vol.470, 1975.

A. Brown and F. Rodriguez-hertz, Measure rigidity for random dynamics on surfaces and related skew products, J. Amer. Math. Soc, vol.30, pp.1055-1132, 2017.

K. Burns, D. Dolgopyat, and Y. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Statist. Phys, vol.108, pp.927-942, 2002.

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math, vol.171, pp.451-489, 2010.

C. Cheng, S. Gan, and Y. Shi, A robustly transitive diffeomorphism of Kan's type, Disc. and Cont. Dyn. Sys, vol.38, pp.867-888, 2018.

B. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep, vol.52, pp.264-379, 1979.

V. Climenhaga, D. Dolgopyat, and Y. Pesin, Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps, Comm. Math. Phys, vol.346, pp.553-602, 2016.

V. Climenhaga, S. Luzzatto, and Y. Pesin, SRB measures and Young towers for surface diffeomorphisms, 2019.

V. Climenhaga, S. Luzzatto, and Y. Pesin, The geometric approach for constructing Sinai-Ruelle-Bowen measures, J. Stat. Phys, vol.166, pp.467-493, 2017.

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, vol.226, pp.673-726, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00325510

S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math, vol.201, pp.385-517, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00538111

L. Díaz, E. Pujals, and R. Ures, Partial hyperbolicity and robust transitivity, Acta Mathematica, vol.183, pp.1-43, 1999.

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.11, pp.359-409, 1994.

M. Grayson, C. Pugh, and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math, vol.140, pp.295-329, 1994.

A. Gorodetski, On stochastic sea of the standard map, Comm. Math. Phys, vol.309, pp.155-192, 2012.

V. Horita and M. Sambarino, Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves, Comment. Math. Helv, vol.92, pp.467-512, 2017.

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, vol.91, pp.216-304, 1939.

F. Izraelev, Nearly linear mappings and their applications, Physica D: Nonlinear Phenomena, vol.1, pp.243-266, 1980.

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, vol.51, pp.137-173, 1980.

A. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), vol.98, pp.527-530, 1954.

A. Kolmogorov, Théorie générale des systèmes dynamiques et mécanique classique, Proceedings of the International Congress of Mathematicians, vol.1, pp.315-333, 1954.

A. Koropecki and M. Nassiri, Transitivity of generic semigroups of area-preserving surface diffeomorphisms, Math. Z, vol.266, pp.707-718, 2010.

F. Ledrappier, Propriétés ergodiques des mesures de, Sinaï. Inst. Hautes Études Sci. Publ. Math, vol.59, pp.163-188, 1984.

C. Liang, K. Marin, and J. Yang, Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.35, pp.1687-1706, 2018.

C. Liang and Y. Yang, the C 1 density of nonuniform hyperbolicity in C r conservative diffeomorphisms, Proc. Amer. Math. Soc, vol.145, pp.1539-1552, 2017.

R. Mañé, Contributions to the stability conjecture. Topology, vol.17, pp.383-396, 1978.

R. Mañé, The Lyapunov exponents of generic area preserving diffeomorphisms, International Conference on Dynamical Systems, pp.110-119, 1995.

. Longman, , 1996.

K. Marin, C r -density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms, Comment. Math. Helv, vol.91, pp.357-396, 2016.

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl, 1962.

G. Núñez and J. Rodriguez-hertz, Minimality and stable bernouliness in dimension 3, 2019.

J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math, vol.42, pp.874-920, 1941.

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk, vol.32, pp.55-112, 1977.

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta math, vol.13, pp.1-270, 1890.

R. Potrie, Partial hyperbolicity and attracting regions in 3-dimensional manifolds, 2012.

R. Potrie, Partially hyperbolic diffeomorphisms with a trapping property, Discrete Contin. Dyn. Syst, vol.35, pp.5037-5054, 2015.

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc, vol.2, pp.1-52, 2000.

C. Pugh and M. Shub, Stable ergodicity and stable accessibility. Differential equations and applications, pp.258-268, 1997.

E. Pujals and M. Sambarino, A sufficient condition for robustly minimal foliations, Ergodic Theory Dynam. Systems, vol.26, pp.281-289, 2006.

F. Rodriguez-hertz, J. R. Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math, vol.172, pp.353-381, 2008.

F. Rodriguez-hertz, J. Rodriguez, A. Hertz, R. Tahzibi, and . Ures, New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J, vol.160, pp.599-629, 2011.

D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math, vol.98, pp.619-654, 1976.

D. Shepelyansky and A. Stone, Chaotic landau level mixing in classical and quantum wells, Physical review letters, vol.74, 1995.

Y. Shi, Perturbation of partially hyperbolic automorphisms on Heisenberg nilmanifolds and holonomy maps. Thesis. Available in, 2014.

M. Shub, Topological transitive diffeomorphism on T 4 . Lecture notes in Mathematics 206, 1971.

J. Sina?, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, vol.27, pp.21-64, 1972.

J. Sina?, Topics in ergodic theory, Princeton Mathematical Series, vol.44, 1994.

S. Smale, Differentiable dynamical systems, Bull. Am. Math. Soc, vol.73, pp.747-817, 1967.

A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math, vol.142, pp.315-344, 2004.

M. Viana, Multidimensional nonhyperbolic attractors. Inst. Hautes Études Sci. Publ. Math, vol.85, pp.63-96, 1997.

J. Yang, Entropy along expanding foliations, 2016.

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math, vol.147, pp.585-650, 1998.

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov, vol.90, 1967.

D. Anosov and Y. Sina?, Certain smooth ergodic systems, Uspehi Mat. Nauk, vol.22, pp.107-172, 1967.

A. Avila, S. Crovisier, and A. Wilkinson, C 1 density of stable ergodicity, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02367511

L. Barreira and Y. Pesin, Lyapunov exponents and smooth ergodic theory, 2002.

P. Berger and P. Carrasco, Non-uniformly hyperbolic diffeomorphisms derived from the standard map, Comm. Math. Phys, vol.329, pp.239-262, 2014.

A. Blumenthal, J. Xue, and L. S. Young, Lyapunov exponents for random perturbations of some area-preserving maps including the standard map, Ann. of Math, vol.185, pp.285-310, 2017.

C. Bonatti, L. Díaz, and M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol.102, 2005.

M. Brin and Y. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat, vol.38, pp.170-212, 1974.

A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the C 2 hypothesis in Pugh-Shub and Ledrappier-Young theory, 2016.

K. Burns, D. Dolgopyat, and Y. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, vol.108, pp.927-942, 2002.

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math, vol.171, pp.451-489, 2010.

B. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep, vol.52, pp.264-379, 1979.

S. Crovisier and E. Pujals, Strongly dissipative surface diffeomorphisms, Comment. Math. Helv, vol.93, pp.377-400, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02367498

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.11, pp.359-409, 1994.

A. Gorodetski, On stochastic sea of the standard map, Comm. Math. Phys, vol.309, pp.155-192, 2012.

M. Grayson, C. Pugh, and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math, vol.140, pp.295-329, 1994.

F. Rodriguez-hertz, J. Rodriguez, A. Hertz, R. Tahzibi, and . Ures, New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J, vol.160, pp.599-629, 2011.

M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, vol.583, 1977.

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, 1939.

V. Horita and M. Sambarino, Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves, Comment. Math. Helv, vol.92, pp.467-512, 2017.

F. Izraelev, Nearly linear mappings and their applications, Physica D: Nonlinear Phenomena, vol.1, pp.243-266, 1980.

D. Obata, On the holonomies of strong stable foliations. Notes on D. Obata's personal webpage, 2018.

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk, vol.32, pp.55-112, 1977.

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc, vol.2, pp.1-52, 2000.

C. Pugh, M. Shub, and A. Wilkinson, Hölder foliations, Duke Math. J, vol.86, pp.517-546, 1997.

C. Pugh, M. Shub, and A. Wilkinson, Correction to: "Hölder foliations, Duke Math. J, vol.105, pp.105-106, 2000.

C. Pugh, M. Shub, and A. Wilkinson, Hölder foliations, revisited, J. Mod. Dyn, vol.6, pp.79-120, 2012.

C. Pugh, M. Viana, and A. Wilkinson, Absolute continuity of foliations. Preprint in M. Viana's personal webpage, 2007.

V. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952.

D. Shepelyansky and A. Stone, Chaotic landau level mixing in classical and quantum wells, Physical review letters, vol.74, 1995.

Y. Sina?, Topics in ergodic theory, Princeton Mathematical Series, vol.44, 1994.

M. Viana, Multidimensional nonhyperbolic attractors. Inst. Hautes Études Sci. Publ. Math, vol.85, pp.63-96, 1997.

F. Abdenur, C. Bonatti, and S. Crovisier, Nonuniform hyperbolicity for C 1 -generic diffeomorphisms, Israel J. Math, vol.183, pp.1-60, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00322876

J. Alves, C. Bonatti, and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Inventiones mathematicae, vol.140, pp.351-398, 2000.

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov, vol.90, 1967.

D. Anosov and Y. Sina?, Certain smooth ergodic systems, Uspehi Mat. Nauk, vol.22, pp.107-172, 1967.

A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory and Dynamical Systems, vol.27, pp.1399-1417, 2007.

A. Arbieto, C. Matheus, and M. J. Pacífico, The Bernoulli property for weakly hyperbolic systems, J. Statist. Phys, vol.117, pp.243-260, 2004.

A. Avila, On the regularization of conservative maps, Acta Math, vol.205, pp.5-18, 2010.

A. Avila, S. Crovisier, and A. Wilkinson, C 1 density of stable ergodicity, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02367511

A. Avila, S. Crovisier, and A. Wilkinson, Diffeomorphisms with positive metric entropy, Publ. Math. Inst. Hautes Études Sci, vol.124, pp.319-347, 2016.

L. Barreira and Y. Pesin, Lyapunov exponents and smooth ergodic theory, 2002.

C. Bonatti and S. Crovisier, Récurrence and généricité, C. R. Math. Acad. Sci, vol.336, pp.839-844, 2003.

C. Bonatti, L. Díaz, and E. Pujals, A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math, vol.158, pp.355-418, 2003.

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math, vol.115, pp.157-193, 2000.

M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Prilozen, vol.1, pp.9-19, 1975.

M. Brin and Y. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat, vol.38, pp.170-212, 1974.

K. Burns, D. Dolgopyat, and Y. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, Journal of Statistical Physics, vol.108, pp.927-942, 2002.

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math, vol.171, pp.451-489, 2010.

J. Buzzi and T. Fisher, Entropic stability beyond partial hyperbolicity, J. Mod. Dyn, vol.4, pp.527-552
URL : https://hal.archives-ouvertes.fr/hal-00628078

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, vol.226, pp.673-726, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00325510

S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Inventiones mathematicae, vol.201, pp.385-517, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00538111

M. Grayson, C. Pugh, and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math, vol.140, pp.295-329, 1994.

F. R. Hertz, M. R. Hertz, A. Tahzibi, and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity, Duke Math. J, vol.160, pp.599-629, 2011.

M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, vol.583, 1977.

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, 1939.

R. Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.8, issue.3

. Springer-verlag, , 1987.

J. Palis and W. De-melo, Geometric theory of dynamical systems, 1982.

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk, vol.32, pp.55-112, 1977.

R. Potrie, Partially hyperbolic diffeomorphisms with a trapping property, Discrete Contin. Dyn. Syst, vol.35, pp.5037-5054, 2015.

C. Pugh and M. Shub, Stable ergodicity and stable accessibility, Differential equations and applications, pp.258-268, 1996.

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc, vol.2, pp.1-52, 2000.

A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math, vol.142, pp.315-344, 2004.

A. Avila, On the regularization of conservative maps, Acta Mathematica, vol.205, pp.5-18, 2010.

A. Avila, J. Santamaria, and M. Viana, Holonomy invariance: rough regularity and applications to Lyapunov exponents, Astérisque, vol.358, pp.13-74, 2013.

A. Avila and M. Viana, On stable accessibility with 2-dimensional center, Preprint in M. Viana's personal webpage

A. Avila and M. Viana, Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math, vol.181, issue.1, pp.115-189, 2010.

L. Barreira and Y. Pesin, Lectures on Lyapunov exponents and smooth ergodic theory, Smooth ergodic theory and its applications, vol.69, pp.3-106, 1999.

P. G. Barrientos and D. Malicet, Extremal exponents of random products of conservative diffeomorphisms, 2018.

P. Berger and D. Turaev, On Herman's positive entropy conjecture, 2017.

J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. & Dynam. Sys, vol.22, pp.1667-1696, 2002.

R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math, vol.92, pp.725-747, 1970.

R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math, vol.470, 1975.

A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the C 2 hypothesis in pugh-shub and ledrappier-young theory, 2016.

A. Brown and F. Hertz, Measure rigidity for random dynamics on surfaces and related skew products, Journal of the American Mathematical Society, vol.30, issue.4, pp.1055-1132, 2017.

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math, vol.171, pp.451-489, 2010.

A. Cannas and . Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol.1764, 2001.

V. Horita and M. Sambarino, Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves, Comment. Math. Helv, vol.92, issue.3, pp.467-512, 2017.

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms, Publ. Math. IHES, vol.51, pp.137-173, 1980.

A. Koropecki and M. Nassiri, Transitivity of generic semigroups of area-preserving surface diffeomorphisms, Mathematische Zeitschrift, vol.266, issue.3, pp.707-718, 2010.

F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, Lyapunov exponents, vol.1186, pp.56-73, 1984.

C. Liang, K. Marin, and J. Yang, C 1 -openness of non-uniform hyperbolic diffeomorphisms with bounded C 2 norm, 2018.

C. Liang and Y. Yang, the c 1 density of nonuniform hyperbolicity in c r conservative diffeomorphisms, Proc. Amer. Math. Soc, vol.145, pp.1539-1552, 2017.

R. Mañé, The Lyapunov exponents of generic area preserving diffeomorphisms, International Conference on Dynamical Systems, pp.110-119, 1995.

K. Marin, C r -density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms, Comment. Math. Helv, vol.91, issue.2, pp.357-396, 2016.

.. B. Ya and . Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, vol.324, pp.55-114, 1977.

M. Poletti, Stably positive lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms, Discrete & Contin. Dyn. Syst, vol.38, 2018.

S. Smale, Differentiable dynamical systems, Bull. Am. Math. Soc, vol.73, pp.747-817, 1967.

A. Tahzibi and J. Yang, Invariance principle and rigidity of high entropy measures, Transactions of American Mathematica Society, 2018.

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math, vol.167, pp.643-680, 2008.

M. Viana and K. Oliveira, Foundations of Ergodic Theory, 2015.

-. Icex and A. , Presidente Antônio Carlos 6627, pp.31270-901

, An f invariant closed set ? admits an dominated splitting if T?M = E ? F is an Df -invariant decomposition satisfying Df n 0 (m)|E(m) · Df ?n 0 |F (f n 0 m) ? 1 2 for some uniform n0 ? 1

P. Berger and P. Carrasco, Non-uniformly hyperbolic diffeomorphisms derived from the standard map, Comm. Math. Phys, vol.329, pp.239-262, 2014.

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Annals of Mathematics, vol.143, issue.2, p.357, 1996.

C. Bonatti, L. Díaz, and E. Pujals, A C 1 -generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Mathematics, vol.158, issue.2, pp.355-418, 2003.

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics, vol.115, issue.1, pp.157-193, 2000.

C. Cheng, S. Gan, and Y. Shi, A robustly transitive diffeomorphism of Kan's type, Disc. and Cont. Dyn. Sys, vol.38, issue.2, pp.867-888, 2018.

L. Díaz, E. Pujals, and R. Ures, Partial hyperbolicity and robust transitivity, Acta Mathematica, vol.183, issue.1, pp.1-43, 1999.

M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, vol.583, 1977.

R. Mañé, Contributions to the stability conjecture, Topology, vol.17, issue.4, pp.383-396, 1978.

D. Obata, On the stable ergodicity of Berger-Carrasco's example. Ergodic Theory and Dynamical Systems, Online version, 2018.

R. Potrie, Partial hyperbolicity and attracting regions in 3-dimensional manifolds, 2012.

E. Pujals and M. Sambarino, A sufficient condition for robustly minimal foliations, Ergodic Theory and Dynamical Systems, vol.26, issue.1, pp.281-289, 2006.

M. Shub, Topological transitive diffeomorphism on T 4 . Lecture notes in Mathematics 206, 1971.

S. Smale, Differentiable Dynamical Systems, Bull. Amer. Math. Soc, vol.73, issue.73, pp.747-817, 1967.

J. Yang, Entropy along expanding foliations, 2016.

J. Alves, C. Bonatti, and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math, vol.140, pp.351-398, 2000.

A. Avila and M. Viana, Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math, vol.181, pp.115-189, 2010.

L. Barreira and Y. Pesin, Lyapunov exponents and smooth ergodic theory, 2002.

S. and B. Ovadia, Generalized SRB measures, physical properties, and thermodynamic formalism of smooth hyperbolic systems, 2019.

P. Berger and P. Carrasco, Non-uniformly hyperbolic diffeomorphisms derived from the standard map, Comm. Math. Phys, vol.329, pp.239-262, 2014.

A. Blumenthal, J. Xue, and L. S. Young, Lyapunov exponents for random perturbations of some area-preserving maps including the standard map, Ann. of Math, vol.185, pp.285-310, 2017.

C. Bonatti, L. Díaz, and M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol.102, 2005.

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math, vol.115, pp.157-193, 2000.

R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, vol.470, 1975.

A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the C 2 hypothesis in Pugh-Shub and Ledrappier-Young theory, 2016.

A. Brown and F. Rodriguez-hertz, Measure rigidity for random dynamics on surfaces and related skew products, J. Amer. Math. Soc, vol.30, issue.4, pp.1055-1132, 2017.

P. Carrasco and D. Obata, A new example of robustly transitive diffeomorphism, 2019.

B. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep, vol.52, pp.264-379, 1979.

V. Climenhaga, D. Dolgopyat, and Y. Pesin, Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps, Comm. Math. Phys, vol.346, pp.553-602, 2016.

V. Climenhaga, S. Luzzatto, and Y. Pesin, SRB measures and Young towers for surface diffeomorphisms, 2019.

V. Climenhaga, S. Luzzatto, and Y. Pesin, The geometric approach for constructing Sinai-Ruelle-Bowen measures, J. Stat. Phys, vol.166, pp.467-493, 2017.

S. Crovisier and E. Pujals, , 2016.

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.11, pp.359-409, 1994.

H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc, vol.108, pp.377-428, 1963.

A. Gorodetski, On stochastic sea of the standard map, Comm. Math. Phys, vol.309, pp.155-192, 2012.

M. Hirayama and N. Sumi, On the ergodicity of hyperbolic Sina?-Ruelle-Bowen measures: the constant unstable dimension case, Ergodic Theory Dynam. Systems, vol.36, pp.1494-1515, 2016.

M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, vol.583, 1977.

F. Izraelev, Nearly linear mappings and their applications, Physica D: Nonlinear Phenomena, vol.1, pp.243-266, 1980.

F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, Lyapunov exponents, vol.1186, pp.56-73, 1984.

F. Ledrappier, Propriétés ergodiques des mesures de, Sinaï. Inst. Hautes Études Sci. Publ. Math, vol.59, pp.163-188, 1984.

F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math, vol.122, pp.509-539, 1985.