Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents
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- by Stefano Luzzatto and Fernando J. Sánchez-Salas PDF
- Proc. Amer. Math. Soc. 141 (2013), 3157-3169 Request permission
Abstract:
We show that for any $C^{1+\alpha }$ diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence $\Omega _{n}$ of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence $\{\mu _{n}\}$ of $f$-invariant ergodic probability measures with $supp (\mu _{n}) \subseteq \Omega _{n}$ we have $\mu _{n}\to \mu$ in the weak-$*$ topology.References
- Luis Barreira and Yakov Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR 2348606, DOI 10.1017/CBO9781107326026
- Xiongping Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity 23 (2010), no. 3, 513–528. MR 2586367, DOI 10.1088/0951-7715/23/3/004
- Dmitry Dolgopyat and Yakov Pesin, Every compact manifold carries a completely hyperbolic diffeomorphism, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 409–435. MR 1898798, DOI 10.1017/S0143385702000202
- Katrin Gelfert, Repellers for non-uniformly expanding maps with singular or critical points, Bull. Braz. Math. Soc. (N.S.) 41 (2010), no. 2, 237–257. MR 2738913, DOI 10.1007/s00574-010-0012-1
- Boris Hasselblatt, Hyperbolic dynamical systems, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 239–319. MR 1928520, DOI 10.1016/S1874-575X(02)80005-4
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- Michihiro Hirayama, Periodic probability measures are dense in the set of invariant measures, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1185–1192. MR 1974422, DOI 10.3934/dcds.2003.9.1185
- A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822, DOI 10.1007/BF02684777
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- Chao Liang, Geng Liu, and Wenxiang Sun, Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1543–1579. MR 2457408, DOI 10.1090/S0002-9947-08-04630-8
- Leonardo Mendoza, Ergodic attractors for diffeomorphisms of surfaces, J. London Math. Soc. (2) 37 (1988), no. 2, 362–374. MR 928529, DOI 10.1112/jlms/s2-37.2.362
- Sheldon E. Newhouse, Lectures on dynamical systems, Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978) Progr. Math., vol. 8, Birkhäuser, Boston, Mass., 1980, pp. 1–114. MR 589590
- Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440 (Russian). MR 0458490
- Clark Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. Stability, symbolic dynamics, and chaos. MR 1396532
- Fernando José Sánchez-Salas, Ergodic attractors as limits of hyperbolic horseshoes, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 571–589. MR 1898806, DOI 10.1017/S0143385702000287
- Ilie Ugarcovici, On hyperbolic measures and periodic orbits, Discrete Contin. Dyn. Syst. 16 (2006), no. 2, 505–512. MR 2226494, DOI 10.3934/dcds.2006.16.505
- Zhenqi Wang and Wenxiang Sun, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4267–4282. MR 2608406, DOI 10.1090/S0002-9947-10-04947-0
- Stephen Wiggins, Global bifurcations and chaos, Applied Mathematical Sciences, vol. 73, Springer-Verlag, New York, 1988. Analytical methods. MR 956468, DOI 10.1007/978-1-4612-1042-9
- Katrin Gelfert and Christian Wolf, On the distribution of periodic orbits, Discrete Contin. Dyn. Syst. 26 (2010), no. 3, 949–966. MR 2600724, DOI 10.3934/dcds.2010.26.949
Additional Information
- Stefano Luzzatto
- Affiliation: Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- Email: luzzatto@ictp.it
- Fernando J. Sánchez-Salas
- Affiliation: Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad del Zulia, Avenida Universidad, Edificio Grano de Oro, Maracaibo, Venezuela
- Email: fjss@fec.luz.edu.ve
- Received by editor(s): June 7, 2011
- Received by editor(s) in revised form: July 14, 2011, and November 25, 2011
- Published electronically: May 24, 2013
- Additional Notes: Most of this work was carried out at the Abdus Salam International Centre for Theoretical Physics (ICTP). The second author was partially supported by the Associateship Programme of ICTP
- Communicated by: Bryna Kra
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3157-3169
- MSC (2010): Primary 37D25
- DOI: https://doi.org/10.1090/S0002-9939-2013-11565-0
- MathSciNet review: 3068969