\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Infimum of the metric entropy of volume preserving Anosov systems

  • * Corresponding author: Miaohua Jiang

    * Corresponding author: Miaohua Jiang 

Y. Jiang is partially supported by the collaboration grant from the Simons Foundation [grant number 199837] and awards from PSC-CUNY and grants from NSFC [grant numbers 11171121 and 11571122]

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we continue our study [9] of the infimum of the metric entropy of the SRB measure in the space of hyperbolic dynamical systems on a smooth Riemannian manifold of higher dimension. We restrict our study to the space of volume preserving Anosov diffeomorphisms and the space of volume preserving expanding endomorphisms. In our previous paper, we use the perturbation method at a hyperbolic periodic point. It raises the question whether the volume can be preserved. In this paper, we answer this question affirmatively. We first construct a smooth path starting from any point in the space of volume preserving Anosov diffeomorphisms such that the metric entropy tends to zero as the path approaches the boundary of the space. Similarly, we construct a smooth path starting from any point in the space of volume preserving expanding endomorphisms with a fixed degree greater than one such that the metric entropy tends to zero as the path approaches the boundary of the space. Therefore, the infimum of the metric entropy as a functional is zero in both spaces.

    Mathematics Subject Classification: Primary: 37C40; Secondary: 37A35, 37D20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems. With an appendix by David Diica and Yakov Simpson-Weller, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417.  doi: 10.1017/S014338570700017X.
    [2] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.  doi: 10.1007/BF01232446.
    [3] J. BochiB. R. Fayad and E. Pujals, A remark on conservative diffeomorphisms, C. R. Math. Acad. Sci. Paris, 342 (2006), 763-766.  doi: 10.1016/j.crma.2006.03.028.
    [4] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Mathematics, 470 Springer-Verlag, New York, 1975.
    [5] B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, (French summary), Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1-26.  doi: 10.1016/S0294-1449(16)30307-9.
    [6] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583 Springer-Verlag, Berlin-New York, 1977.
    [7] H. Hu, Conditions for the existence of SBR measures of ''almost Anosov'' diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367.  doi: 10.1090/S0002-9947-99-02477-0.
    [8] H. Hu, Statistical properties of some almost hyperbolic systems, in Smooth Ergodic Theory and Its Applications(Seattle, 1999), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 69 (2001), 367–384. doi: 10.1090/pspum/069/1858539.
    [9] H. HuM. Jiang and Y. Jiang, Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure, Discrete Contin. Dyn. Syst., 22 (2008), 215-234.  doi: 10.3934/dcds.2008.22.215.
    [10] H. Hu and L.-S. Young, Nonexistence of SBR measures for some diffeomorphisms that are ''almost Anosov'', Ergod. Th. & Dynam. Sys., 15 (1995), 67-76.  doi: 10.1017/S0143385700008245.
    [11] M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors, Ergod. Th. & Dynam. Sys., 32 (2012), 1350-1369.  doi: 10.1017/S0143385711000241.
    [12] Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, arXiv: 0804.3104v3
    [13] A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math., 110 (1979), 529-547.  doi: 10.2307/1971237.
    [14] A. Katok and B. Hasselbratt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.
    [15] F. Ledrappier, Propriétés ergodiques des measure de Sinai, Publ. Math. I.H.E.S., 59 (1984), 163-188. 
    [16] F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergod. Th. & Dynam. Sys., 2 (1982), 203-219.  doi: 10.1017/S0143385700001528.
    [17] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms Ⅰ, Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539.  doi: 10.2307/1971328.
    [18] W. LiJ. Llibre and X. Zhang, Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows, Amer. J. Math., 124 (2002), 107-127.  doi: 10.1353/ajm.2002.0004.
    [19] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.  doi: 10.1090/S0002-9947-1965-0182927-5.
    [20] V. A. Rokhlin, Lectures on the theory of entropy of transformations with invariant measures, Uspehi Mat. Nauk, 22 (1967), 3-56. 
    [21] D. Ruelle, Differentiation of SRB states, Commun. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.
    [22] Ya. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64. 
  • 加载中
SHARE

Article Metrics

HTML views(139) PDF downloads(222) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return