Hostname: page-component-857557d7f7-d5hhr Total loading time: 0 Render date: 2025-11-21T12:43:43.065Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-uniformly hyperbolic endomorphisms

Part of: Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  01 September 2025

Martin Andersson Open the ORCID record for Martin Andersson [Opens in a new window] ,
Pablo D. Carrasco Open the ORCID record for Pablo D. Carrasco [Opens in a new window]  and
Radu Saghin Open the ORCID record for Radu Saghin [Opens in a new window]
Martin Andersson
Affiliation:
Universidade Federal Fluminense, Departamento de Matemática aplicada, Rua Professor Marcos Waldemar de Freitas Reis, 24210-201, Niterói, RJ, Brazil nilsmartin@id.uff.br
Pablo D. Carrasco
Affiliation:
Universidade Federal de Minas Gerais, ICEx-UFMG, Av. Pres. Antônio Carlos, 6627 - 31270-901, Belo Horizonte, MG, Brazil pdcarrasco@gmail.com
Radu Saghin
Affiliation:
Pontificia Universidad Católica de Valparaíso, Instituto de Matemática, Blanco Viel 596, 2350050, Cerro Barón, Valparaíso, Chile rsaghin@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We show the existence of large $\mathcal C^1$ open sets of area-preserving endomorphisms of the two-torus which have no dominated splitting and are non-uniformly hyperbolic, meaning that Lebesgue almost every point has a positive and a negative Lyapunov exponent. The integrated Lyapunov exponents vary continuously with the dynamics in the $\mathcal C^1$ topology and can be taken as far away from zero as desired. Explicit real analytic examples are obtained by deforming linear endomorphisms, including expanding ones. The technique works in nearly every homotopy class, and the examples are stably ergodic (in fact Bernoulli), provided that the linear map has no eigenvalue of modulus one.

Keywords

non-uniform hyperbolicityLyapunov exponentsstable ergodicity

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A05: Measure-preserving transformations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 6 , June 2025 , pp. 1313 - 1356
Check for updates
Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alves, J., Bonatti, C. and Viana, M., SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351–398.10.1007/s002220000057CrossRefGoogle Scholar
Andersson, M., Robust ergodic properties in partially hyperbolic dynamics, Trans. Amer. Math. Soc. 362 (2010), 1831–1867.10.1090/S0002-9947-09-05027-2CrossRefGoogle Scholar
Andersson, M., Transitivity of conservative toral endomorphisms, Nonlinearity 29 (2016), 1047–1055.10.1088/0951-7715/29/3/1047CrossRefGoogle Scholar
Avila, A., Crovisier, S. and Wilkinson, A., $\mathcal {C}^1$ density of stable ergodicity, Adv. Math. 379 (2021), 107496.10.1016/j.aim.2020.107496CrossRefGoogle Scholar
Barreira, L. and Pesin, Y., Introduction to smooth ergodic theory (American Mathematical Society, Providence, RI, 2013).Google Scholar
Berger, P. and Carrasco, P. D., Non-uniformly hyperbolic diffeomorphisms derived from the standard map, Comm. Math. Phys. 329 (2014), 239–262.10.1007/s00220-014-1932-1CrossRefGoogle Scholar
Blumenthal, A., Xue, J. and Young, L.-S., Lyapunov exponents for random perturbations of some area-preserving maps including the standard map, Ann. of Math. (2) 185 (2017), 285–310.10.4007/annals.2017.185.1.5CrossRefGoogle Scholar
Bochi, J., Genericity of zero lyapunov exponents, Ergodic Theory Dynam. Systems 22 (2002), 1667–1696.10.1017/S0143385702001165CrossRefGoogle Scholar
Bochi, J., Fayad, B. R. and Pujals, E., A remark on conservative diffeomorphisms, C. R. Math. Acad. Sci. Paris 342 (2006), 763–766.10.1016/j.crma.2006.03.028CrossRefGoogle Scholar
Bochi, J., Katok, A. and Hertz, F. R., Flexibility of lyapunov exponents, Ergodic Theory Dynam. Systems 42 (2022), 554–591.10.1017/etds.2021.78CrossRefGoogle Scholar
Brown, A. and Rodriguez Hertz, F., Measure rigidity for random dynamics on surfaces and related skew products, J. Amer. Math. Soc. 30 (2017), 1055–1132.10.1090/jams/877CrossRefGoogle Scholar
Burns, K. and Wilkinson, A., On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), 451–489.10.4007/annals.2010.171.451CrossRefGoogle Scholar
Carrasco, P. D., Random products of standard maps, Comm. Math. Phys. 377 (2020), 773–810.10.1007/s00220-020-03765-6CrossRefGoogle Scholar
Carrasco, P. D. and Saghin, R., Extended flexibility of lyapunov exponents for Anosov diffeomorphisms, Trans. Amer. Math. Soc. 375 (2022), 3411–3449.Google Scholar
Chirikov, B. and Shepelyansky, D., Chirikov standard map, Scholarpedia 3 (2008), 3550.10.4249/scholarpedia.3550CrossRefGoogle Scholar
Chung, P. N., Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces, Preprint (2020), arXiv:2006.03166.Google Scholar
Dolgopyat, D. and Krikorian, R., On simultaneous linearization of diffeomorphisms of the sphere, Duke. Math. J. 136 (2007), 475–505.10.1215/S0012-7094-07-13633-0CrossRefGoogle Scholar
Dolgopyat, D. and Pesin, Y., Every compact manifold carries a completely hyperbolic diffeomorphism, Ergodic Theory Dynam. Systems 22 (2002), 409–435.10.1017/S0143385702000202CrossRefGoogle Scholar
Furstenberg, H. and Kesten, H., Products of random matrices, Ann. Math. Stat. 31 (1960), 457–469.10.1214/aoms/1177705909CrossRefGoogle Scholar
Grayson, M., Pugh, C. and Shub, M., Stably ergodic diffeomorphisms, Ann. of Math. (2) 140 (1994), 295–329.10.2307/2118602CrossRefGoogle Scholar
Hertz, F. R., Hertz, M. A. R., Tahzibi, A. and Ures, R., Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Comm. Math. Phys. 306 (2011), 35–49.10.1007/s00220-011-1275-0CrossRefGoogle Scholar
Hertz, F. R., Hertz, M. A. R. and Ures, R., Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), 353–381.CrossRefGoogle Scholar
Jacobson, N., Basic algebra I (Dover Publications, 2009).Google Scholar
Janeiro, V., Existence of robust non-uniformly hyperbolic endomorphism in homotopy classes, J. Dyn. Control Syst. 29 (2023), 2023–2039.10.1007/s10883-023-09668-8CrossRefGoogle Scholar
Katok, A., Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2) 2 (1979), 529–547.10.2307/1971237CrossRefGoogle Scholar
Ledrappier, F., Propriétés ergodiques des mesures de sinaï, Publ. Math. Inst. Hautes Études Sci. 59 (1984), 163–188.CrossRefGoogle Scholar
Ledrappier, F. and Young, L.-S., The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), 509–539.CrossRefGoogle Scholar
Liang, C., Marin, K. and Yang, J., Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), 1687–1706.10.1016/j.anihpc.2018.01.007CrossRefGoogle Scholar
Lima, Y., Obata, D. and Poletti, M., Measures of maximal entropy for non-uniformly hyperbolic maps, Preprint (2024), arXiv:2405.04676.Google Scholar
Liu, P.-D., Invariant measures satisfying an equality relating entropy, folding entropy and negative lyapunov exponents, Comm. Math. Phys. 284 (2008), 391–406.10.1007/s00220-008-0568-4CrossRefGoogle Scholar
Liu, X., Lyapunov exponents approximation, symplectic cocycle deformation and alarge deviation theorem, Thesis, IMPA (2016).Google Scholar
Liu, P. D. and Shu, L., Absolute continuity of hyperbolic invariant measures for endomorphisms, Nonlinearity 24 (2011), 1595–1611.10.1088/0951-7715/24/5/011CrossRefGoogle Scholar
Moore, C. C. and Schochet, C., Global analysis on foliated spaces (Springer, New York, 2012).Google Scholar
Obata, D., On the stable ergodicity of Berger--Carrasco’s example, Ergodic Theory Dynam. Systems 40 (2020), 1008–1056.10.1017/etds.2018.65CrossRefGoogle Scholar
Oseledets, V. I., A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Pugh, C. and Shub, M., Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), 1–54.10.1090/S0002-9947-1989-0983869-1CrossRefGoogle Scholar
Qian, M. and Zhu, S., SRB measures and Pesin’s entropy formula for endomorphisms, Trans. Amer. Math. Soc. 354 (2002), 1453–1471.10.1090/S0002-9947-01-02792-1CrossRefGoogle Scholar
Ramírez, S. and Vivas, K., Non-uniform hyperbolicity of maps on $\mathbb {T}^2$ , Preprint (2023), arXiv:2312.16742.Google Scholar
Rotman, J., An introduction to algebraic topology, Graduate Texts in Mathematics, vol. 119 (Springer, New York, 1988).CrossRefGoogle Scholar
Ruelle, D., Ergodic theory of differentiable dynamical systems, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 27–58.10.1007/BF02684768CrossRefGoogle Scholar
Shub, M., Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199.10.2307/2373276CrossRefGoogle Scholar
Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc 73 (1967), 747–817.10.1090/S0002-9904-1967-11798-1CrossRefGoogle Scholar
Tsujii, M., Physical measures for partially hyperbolic surface endomorphisms, Acta Math. 194 (2005), 37–132.CrossRefGoogle Scholar
Tsujii, M. and Zhang, Z., On bifurcation of statistical properties of partially hyperbolic endomorphisms, Ergodic Theory Dynam. Systems 44 (2024), 933–944.10.1017/etds.2023.11CrossRefGoogle Scholar
Verjovsky, A., Low-dimensional solenoidal manifolds, EMS Surv. Math. Sci. 10 (2023), 131–178.10.4171/emss/69CrossRefGoogle Scholar
Viana, M., Lectures on Lyapunov exponents (Cambridge University Press, 2014).10.1017/CBO9781139976602CrossRefGoogle Scholar
Viana, M., (Dis)continuity of Lyapunov exponents, Ergodic Theory Dynam. Systems 40 (2018), 577–611.10.1017/etds.2018.50CrossRefGoogle Scholar
Viana, M. and Yang, J., Continuity of Lyapunov exponents in the $\mathcal C^0$ topology, Israel J. Math. 229 (2019), 461–485.10.1007/s11856-018-1809-7CrossRefGoogle Scholar
Williams, R., Expanding attractors, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 169–203.10.1007/BF02684369CrossRefGoogle Scholar
Yang, J., Entropy along expanding foliations, Adv. Math. 389 (2021), 107893.10.1016/j.aim.2021.107893CrossRefGoogle Scholar