SRB measures for pointwise hyperbolic systems on open regions. (English) Zbl 1450.37020
Summary: A diffeomorphism \(f:M\rightarrow M\) is pointwise partially hyperbolic on an open invariant subset \(N\subset M\) if there is an invariant decomposition \(T_NM = E^{u}\oplus E^c\oplus E^s\) such that \(D_xf\) is strictly expanding on \(E_x^{u}\) and contracting on \(E_x^{u}\) at each \(x\in N\). We show that under certain conditions \(f\) has unstable and stable manifolds, and admits a finite or an infinite \(u\)-Gibbs measure \(\mu\). If \(f\) is pointwise hyperbolic on \(N\), then \(\mu\) is a Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.
MSC:
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37D30 | Partially hyperbolic systems and dominated splittings |
| 37D10 | Invariant manifold theory for dynamical systems |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
Keywords:
pointwise hyperbolicity; unstable manifold; SRB measure; almost Anosov diffeomorphism; Katok’s mapReferences:
| [1] | Aaronson, J., An Introduction to Infinite Ergodic Theory (1997), Providence: Amer Math Soc, Providence · Zbl 0882.28013 |
| [2] | Alves, J.; Leplaideur, R., SRB measures for almost Axiom A diffeomorphisms, Ergodic Theory Dynam Systems, 36, 2015-2043 (2016) · Zbl 1370.37046 · doi:10.1017/etds.2015.4 |
| [3] | Barreira L, Pesin Y. Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series, vol. 23. Providence: Amer Math Soc, 2002 |
| [4] | Brin, M., Bernoulli diffeomorphisms with n−1 non-zero exponents, Ergodic Theory Dynam Systems, 1, 1-7 (1981) · Zbl 0485.58011 · doi:10.1017/S0143385700001127 |
| [5] | Brin, M.; Feldman, J.; Katok, A., Bernoulli diffeomorphisms and group extensions of dynamical systems with non-zero characteristic exponents, Ann of Math (2), 113, 159-179 (1981) · Zbl 0477.58021 · doi:10.2307/1971136 |
| [6] | Bruin, H.; Terhesiu, D., Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms, Ergodic Theory Dynam Systems, 40, 663-698 (2020) · Zbl 1437.37003 · doi:10.1017/etds.2018.58 |
| [7] | Burns, K.; Wilkinson, A., On the ergodicity of partially hyperbolic systems, Ann of Math (2), 171, 451-489 (2010) · Zbl 1196.37057 · doi:10.4007/annals.2010.171.451 |
| [8] | Catsigeras, E.; Enrich, H., SRB measures of certain almost hyperbolic diffeomorphisms with a tangency, Discrete Contin Dyn Syst, 7, 177-202 (2001) · Zbl 1027.37015 · doi:10.3934/dcds.2001.7.177 |
| [9] | Chen, J.; Hu, H.; Pesin, Y., A volume preserving flow with essential coexistence of zero and non-zero Lyapunov exponents, Ergodic Theory Dynam Systems, 33, 1748-1785 (2013) · Zbl 1286.37037 · doi:10.1017/etds.2012.109 |
| [10] | Chen J, Hu H, Pesin Y, et al. The essential coexistence phenomenon in Hamiltonian dynamics. arXiv:1901.07713, 2019 · Zbl 1287.37007 |
| [11] | Dolgopyat, D.; Pesin, Y., Every compact manifold carries a completely hyperbolic diffeomorphism, Ergodic Theory Dynam Systems, 22, 409-435 (2002) · Zbl 1046.37015 · doi:10.1017/S0143385702000202 |
| [12] | Gerber, M.; Katok, A., Smooth models of Thurston’s pseudo-Anosov maps, Ann Sci Ácole Norm Sup (4), 15, 173-204 (1982) · Zbl 0502.58029 · doi:10.24033/asens.1424 |
| [13] | Hatomoto, J., Diffeomorphisms admitting SRB measures and their regularity, Kodai Math J, 29, 211-226 (2006) · Zbl 1108.37019 · doi:10.2996/kmj/1151936436 |
| [14] | Hu, H., Conditions for the existence of SBR measures for “almost Anosov” diffeomorphisms, Trans Amer Math Soc, 352, 2331-2367 (2000) · Zbl 0949.37007 · doi:10.1090/S0002-9947-99-02477-0 |
| [15] | Hu, H.; Pesin, Y.; Taliskaya, A., Every compact manifold carries a hyperbolic flow, Modern Dynamical Systems and Applications, 347-358 (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1147.37316 |
| [16] | Hu, H.; Pesin, Y.; Taliskaya, A., A volume preserving diffeomorphism with essential coexistence of zero and nonzero Lyapunov exponents, Comm Math Phys, 319, 331-378 (2013) · Zbl 1281.37017 · doi:10.1007/s00220-012-1602-0 |
| [17] | Hu, H.; Young, L-S, Nonexistence of SBR measures for some diffeomorphisms that are “almost Anosov”, Ergodic Theory Dynam Systems, 15, 67-76 (1995) · Zbl 0818.58035 · doi:10.1017/S0143385700008245 |
| [18] | Hu, H.; Zhang, X., Polynomial decay of correlations for almost Anosov diffeomorphisms, Ergodic Theory Dynam Systems, 39, 832-864 (2019) · Zbl 1423.37032 · doi:10.1017/etds.2017.107 |
| [19] | Katok, A., Bernoulli diffeomorphisms on surfaces, Ann of Math (2), 110, 529-547 (1979) · Zbl 0435.58021 · doi:10.2307/1971237 |
| [20] | Leplaideur, R., Existence of SRB-measures for some topologically hyperbolic diffeomorphisms, Ergodic Theory Dynam Systems, 24, 1199-1225 (2004) · Zbl 1062.37012 · doi:10.1017/S0143385704000094 |
| [21] | Pesin, Y.; Senti, S.; Zhang, K., Thermodynamics of the Katok map, Ergodic Theory Dynam Systems, 39, 764-794 (2019) · Zbl 1412.37043 · doi:10.1017/etds.2017.35 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
