Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents

Authors:
Stefano Luzzatto and Fernando J. Sánchez-Salas

Journal:
Proc. Amer. Math. Soc. **141** (2013), 3157-3169

MSC (2010):
Primary 37D25

DOI:
https://doi.org/10.1090/S0002-9939-2013-11565-0

Published electronically:
May 24, 2013

MathSciNet review:
3068969

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Abstract: We show that for any 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 of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence of -invariant ergodic probability measures with we have in the weak- topology.

**1.**L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115. Cambridge University Press, 2007. MR**2348606 (2010c:37067)****2.**X. Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok. Nonlinearity,**23**(2010) 513-528. MR**2586367****3.**D. Dolgopyat and Y. B. Pesin, Every compact manifold carries a completely hyperbolic diffeomorphism. Ergodic Theory and Dynamical Systems,**22(2)**(2002) 409-435. MR**1898798 (2003b:37050)****4.**K. Gelfert, Repellers for non-uniformly expanding maps with singular or critical points, Bulletin of the Brazilian Mathematical Society,**41(2)**(2010) 237-257. MR**2738913****5.**B. Hasselblat, Hyperbolic Dynamical Systems, Handbook of dynamical systems, Vol. 1A, 239-319, North-Holland, Amsterdam, 2002. MR**1928520 (2004b:37047)****6.**B. Hasselblat and A. Katok, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, Cambridge, 1995. MR**1326374 (96c:58055)****7.**M. Hirayama, Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems. Series A,**9(5)**(2003) 1185-1192. MR**1974422 (2004a:37032)****8.**A. Katok, Lyapunov exponents, entropy and periodic orbits of diffeomorphisms. Publ. Math. IHES,**51**(1980) 137-173. MR**573822 (81i:28022)****9.**A. Katok and L. Mendoza, Dynamical systems with non-uniformly hyperbolic behavior. Supplement to Introduction to the Modern Theory of Dynamical Systems, by A. Katok and B. Hasselblat, Encyclopedia of Math. and its Applications, Vol. 54. Cambridge University Press, 1995. MR**1326374 (96c:58055)****10.**C. Liang, G. Liu, and W. Sun, Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems. Transactions of the American Mathematical Society,**361(3)**(2009) 1543-1579. MR**2457408 (2009m:37085)****11.**L. Mendoza, Ergodic attractors for diffeomorphisms of surfaces. Journal of the London Math. Soc.,**37**(1988) 362-374. MR**928529 (89e:58071)****12.**S. Newhouse, Lectures on Dynamical Systems, Dynamical Systems (C.I.M.E. Summer School Bressanone, 1978), pp. 1-114, Progr. Math., 8, Birkhäuser, Boston, Mass., 1980. MR**589590 (81m:58028)****13.**Ya. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Math. USSR Izv.,**40(6)**(1976) 1261-1305. MR**0458490 (56:16690)****14.**C. Robinson, Dynamical Systems: Stability, symbolic dynamics and chaos. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR**1396532 (97e:58064)****15.**Fernando J. Sánchez-Salas, Ergodic attractors as limits of hyperbolic horseshoes, Ergodic Theory and Dynamical Systems,**22**(2002) 571-589. MR**1898806 (2003c:37037)****16.**I. Ugarcovici, On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems. Series A,**16(2)**(2006) 505-512. MR**2226494 (2007c:37025)****17.**Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits, Transactions of the American Mathematical Society,**362(8)**(2010) 4267-4282. MR**2608406 (2011m:37030)****18.**S. Wiggins, Global bifurcations and chaos: analytical methods, Applied Mathematical Sciences, Vol. 73, Springer, New York, 1988. MR**956468 (89m:58057)****19.**C. Wolf and K. Gelfert, On the distribution of periodic orbits. Discrete and Continuous Dynamical Systems, Series A**26(3)**(2010) 949-966. MR**2600724 (2011g:37080)**

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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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11565-0

Keywords:
Non-uniformly hyperbolic systems,
uniformly hyperbolic systems,
approximation of hyperbolic measures

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

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.