Nonhyperbolic step skew-products: Ergodic approximation
Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 6, pp. 1561-1598.

We study transitive step skew-product maps modeled over a complete shift of k, k2, symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents.

We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves.

Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.

DOI: 10.1016/j.anihpc.2016.10.009
Classification: 37D25, 37D35, 37D30, 28D20, 28D99
Keywords: Entropy, Ergodic measures, Lyapunov exponents, Skew-product, Transitivity
@article{AIHPC_2017__34_6_1561_0,
     author = {D{\'\i}az, L.J. and Gelfert, K. and Rams, M.},
     title = {Nonhyperbolic step skew-products: {Ergodic} approximation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1561--1598},
     publisher = {Elsevier},
     volume = {34},
     number = {6},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.10.009},
     zbl = {1475.37044},
     mrnumber = {3712011},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.009/}
}
TY  - JOUR
AU  - Díaz, L.J.
AU  - Gelfert, K.
AU  - Rams, M.
TI  - Nonhyperbolic step skew-products: Ergodic approximation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 1561
EP  - 1598
VL  - 34
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.009/
DO  - 10.1016/j.anihpc.2016.10.009
LA  - en
ID  - AIHPC_2017__34_6_1561_0
ER  - 
%0 Journal Article
%A Díaz, L.J.
%A Gelfert, K.
%A Rams, M.
%T Nonhyperbolic step skew-products: Ergodic approximation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 1561-1598
%V 34
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.009/
%R 10.1016/j.anihpc.2016.10.009
%G en
%F AIHPC_2017__34_6_1561_0
Díaz, L.J.; Gelfert, K.; Rams, M. Nonhyperbolic step skew-products: Ergodic approximation. Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 6, pp. 1561-1598. doi : 10.1016/j.anihpc.2016.10.009. http://www.numdam.org/articles/10.1016/j.anihpc.2016.10.009/

[1] Abdenur, F.; Bonatti, Ch.; Crovisier, S. Nonuniform hyperbolicity for C1-generic diffeomorphisms, Isr. J. Math., Volume 183 (2011), pp. 1–60 | DOI | MR | Zbl

[2] Alves, J.F.; Araújo, V.; Saussol, B. On the uniform hyperbolicity of certain hyperbolic systems, Proc. Am. Math. Soc., Volume 131 (2003), pp. 1303–1309 | MR | Zbl

[3] Bochi, J.; Bonatti, Ch.; Díaz, L.J. Robust vanishing of all Lyapunov exponents for iterated function systems, Math. Z., Volume 276 (2014), pp. 469–503 | DOI | MR | Zbl

[4] Bonatti, Ch.; Díaz, L.J. Persistent nonhyperbolic transitive diffeomorphisms, Ann. Math. (2), Volume 143 (1996), pp. 357–396 | DOI | MR | Zbl

[5] Bonatti, Ch.; Díaz, L.J.; Viana, M. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences – Mathematical Physics, III, vol. 102, Springer, Berlin, 2005 | MR | Zbl

[6] Bonatti, Ch.; Díaz, L.J.; Gorodetski, A. Nonhyperbolic ergodic measures with large support, Nonlinearity, Volume 23 (2010), pp. 687–705 | DOI | MR | Zbl

[7] Bonatti, Ch.; Díaz, L.J.; Ures, R. Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, Volume 1 (2002), pp. 513–541 | DOI | MR | Zbl

[8] Bonatti, Ch.; Gelfert, K. Dominated Pesin theory: convex sums of hyperbolic measures (preprint) | arXiv | Zbl

[9] Bowen, R. Topological entropy for noncompact sets, Trans. Am. Math. Soc., Volume 184 (1973), pp. 125–136 | DOI | MR | Zbl

[10] Brin, M.; Katok, A., Springer Lecture Notes in Mathematics, Volume vol. 1007, Springer-Verlag (1983), pp. 30–38 | DOI | MR | Zbl

[11] Buzzi, J. The almost Borel structure of diffeomorphisms with some hyperbolicity, Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Sympos. Pure Math., vol. 89, Amer. Math. Soc., Providence, RI, 2015, pp. 9–44 | DOI | MR | Zbl

[12] Crovisier, S. Partial hyperbolicity far from homoclinic bifurcations, Adv. Math., Volume 226 (2011), pp. 673–726 | DOI | MR | Zbl

[13] Díaz, L.J.; Gelfert, K.; Rams, M. Abundant rich phase transitions in step-skew products, Nonlinearity, Volume 27 (2014), pp. 2255–2280 | MR | Zbl

[14] Díaz, L.J.; Gorodetski, A. Nonhyperbolic ergodic measures for nonhyperbolic homoclinic classes, Ergod. Theory Dyn. Syst., Volume 29 (2009), pp. 1479–1513 | MR | Zbl

[15] Gelfert, K. Horseshoes for diffeomorphisms preserving hyperbolic measures, Math. Z., Volume 282 (2016), pp. 685–701 | MR | Zbl

[16] Gorodetski, A.; Ilyashenko, Yu.; Kleptsyn, V.; Nalskij, M. Non-removable zero Lyapunov exponent, Funct. Anal. Appl., Volume 39 (2005), pp. 27–38 | DOI | MR | Zbl

[17] Katok, A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES, Volume 51 (1980), pp. 137–173 | DOI | Numdam | MR | Zbl

[18] Katok, A.; Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications, vol. 54, Cambridge University Press, 1995 | MR | Zbl

[19] Kleptsyn, V.; Nal'skii, M.B. Stability of the existence of nonhyperbolic measures for C1-diffeomorphisms, Funkc. Anal. Prilozh., Volume 41 (2007), pp. 30–45 (96; translation in Funct. Anal. Appl., 271–283) | MR | Zbl

[20] Kleptsyn, V.; Volk, D. Physical measures for nonlinear random walks on interval, Mosc. Math. J., Volume 14 (2014), pp. 339–365 | MR | Zbl

[21] Ledrappier, F.; Walters, P. A relativised variational principle for continuous transformations, J. Lond. Math. Soc. (2), Volume 16 (1977), pp. 568–576 | MR | Zbl

[22] Lind, D.; Marcus, B. An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[23] Luzzatto, S.; Sánchez-Salas, F.J. Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents, Proc. Am. Math. Soc., Volume 141 (2013), pp. 3157–3169 | DOI | MR | Zbl

[24] Misiurewicz, M.; Szlenk, W. Entropy of piecewise monotone mappings, Stud. Math., Volume LXVII (1980), pp. 45–63 | MR | Zbl

[25] Pugh, C.; Shub, M. Stably ergodic dynamical systems and partial hyperbolicity, J. Complex., Volume 13 (1997), pp. 125–179 | DOI | MR | Zbl

[26] Ruelle, D.; Wilkinson, A. Absolutely singular dynamical foliations, Commun. Math. Phys., Volume 219 (2001), pp. 481–487 | DOI | MR | Zbl

[27] Rodriguez Hertz, F.; Rodriguez Hertz, M.A.; Ures, R. Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., vol. 51, Amer. Math. Soc., Providence, RI, 2007, pp. 103–109 | MR | Zbl

[28] Rodriguez-Hertz, F.; Rodriguez-Hertz, M.A.; Tahzibi, A.; Ures, R. Maximizing measures for partially hyperbolic systems with compact center leaves, Ergod. Theory Dyn. Syst., Volume 32 (2012), pp. 825–839 | DOI | MR | Zbl

[29] Shub, M.; Wilkinson, A. Pathological foliations and removable zero exponents, Invent. Math., Volume 139 (2000), pp. 495–508 | DOI | MR | Zbl

[30] Schweiger, F. Proceedings of the Sixth Conference on Probability Theory, Academiei Republicii Socialiste Rania (1981), pp. 221–228 (Bucaresti) | MR | Zbl

[31] Walters, P. An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York–Berlin, 1982 | DOI | MR | Zbl

Cited by Sources: