Topological structure of (partially) hyperbolic sets with positive volume

Authors:
José F. Alves and Vilton Pinheiro

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5551-5569

MSC (2000):
Primary 37Dxx

Published electronically:
April 28, 2008

MathSciNet review:
2415085

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is larger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume.

**1.**Flavio Abdenur, Christian Bonatti, and Lorenzo J. Díaz,*Non-wandering sets with non-empty interiors*, Nonlinearity**17**(2004), no. 1, 175–191. MR**2023438**, 10.1088/0951-7715/17/1/011**2.**J. F. Alves, V. Araújo, M. J. Pacifico, V. Pinheiro,*On the volume of singular-hyperbolic sets*, Dyn. Syst., to appear.**3.**José F. Alves, Vítor Araújo, and Benoît Saussol,*On the uniform hyperbolicity of some nonuniformly hyperbolic systems*, Proc. Amer. Math. Soc.**131**(2003), no. 4, 1303–1309. MR**1948124**, 10.1090/S0002-9939-02-06857-0**4.**José F. Alves, Christian Bonatti, and Marcelo Viana,*SRB measures for partially hyperbolic systems whose central direction is mostly expanding*, Invent. Math.**140**(2000), no. 2, 351–398. MR**1757000**, 10.1007/s002220000057**5.**Jairo Bochi and Marcelo Viana,*Lyapunov exponents: how frequently are dynamical systems hyperbolic?*, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 271–297. MR**2090775****6.**Rufus Bowen,*A horseshoe with positive measure*, Invent. Math.**29**(1975), no. 3, 203–204. MR**0380890****7.**Rufus Bowen,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR**0442989****8.**Todd Fisher,*Hyperbolic sets with nonempty interior*, Discrete Contin. Dyn. Syst.**15**(2006), no. 2, 433–446. MR**2199438**, 10.1007/BF02607061**9.**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****10.**Sheldon E. Newhouse,*The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 101–151. MR**556584****11.**J. Palis and F. Takens,*Hyperbolicity and the creation of homoclinic orbits*, Ann. of Math. (2)**125**(1987), no. 2, 337–374. MR**881272**, 10.2307/1971313**12.**Jacob Palis and Jean-Christophe Yoccoz,*Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension*, Acta Math.**172**(1994), no. 1, 91–136. MR**1263999**, 10.1007/BF02392792**13.**Jacob Palis and Jean-Christophe Yoccoz,*Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs*, C. R. Acad. Sci. Paris Sér. I Math.**333**(2001), no. 9, 867–871 (French, with English and French summaries). MR**1873226**, 10.1016/S0764-4442(01)02139-5**14.**Ya. Pesin,*Families of invariant manifolds corresponding to non-zero characteristic exponents*,

Math. USSR Izv.**10**(1976), 1261-1302.**15.**V. A. Pliss,*On a conjecture of Smale*, Differencial′nye Uravnenija**8**(1972), 268–282 (Russian). MR**0299909****16.**Michael Shub,*Global stability of dynamical systems*, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR**869255****17.**Stephen Smale,*Diffeomorphisms with many periodic points*, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR**0182020****18.**Lai-Sang Young,*Large deviations in dynamical systems*, Trans. Amer. Math. Soc.**318**(1990), no. 2, 525–543. MR**975689**, 10.1090/S0002-9947-1990-0975689-7

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37Dxx

Retrieve articles in all journals with MSC (2000): 37Dxx

Additional Information

**José F. Alves**

Affiliation:
Departamento de Matemática Pura, Faculdade de Ciências do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Email:
jfalves@fc.up.pt

**Vilton Pinheiro**

Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil

Email:
viltonj@ufba.br

DOI:
https://doi.org/10.1090/S0002-9947-08-04484-X

Keywords:
Hyperbolic set,
partially hyperbolic set,
horseshoe

Received by editor(s):
June 5, 2006

Received by editor(s) in revised form:
January 8, 2007

Published electronically:
April 28, 2008

Additional Notes:
This work was carried out at the Federal University of Bahia, University of Porto and IMPA. The first author was partially supported by CMUP, by a grant of FCT and by POCI/MAT/61237/2004. The second author was partially supported by PADCT/CNPq and by POCI/MAT/61237/2004

Article copyright:
© Copyright 2008
American Mathematical Society

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