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September  2017, 37(9): 4767-4783. doi: 10.3934/dcds.2017205

Infimum of the metric entropy of volume preserving Anosov systems

Huyi Hu 1, , Miaohua Jiang 2,, and  Yunping Jiang 3,4,

1. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
2. 

Department of Mathematics and Statistics, Wake Forest University, Winston Salem, NC 27109, USA
3. 

Department of Mathematics, Queens College of the City University of New York, Flushing, NY 11367, USA
4. 

Department of Mathematics, Graduate Center of the City University of New York, New York, NY 10016, USA

* Corresponding author: Miaohua Jiang

Received  May 2016 Revised  April 2017 Published  June 2017

Fund Project: 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].

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.

Keywords: Volume preserving Anosov diffeomorphism, volume preserving expanding endomorphism, metric entropy, infimum of metric entropy.
Mathematics Subject Classification: Primary: 37C40; Secondary: 37A35, 37D20.
Citation: Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

J. Bochi, B. 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.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Mathematics, 470 Springer-Verlag, New York, 1975.  Google Scholar

[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.  Google Scholar

[6]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583 Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

H. Hu, M. 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.  Google Scholar

[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.  Google Scholar

[11]

M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors, Ergod. Th. & Dynam. Sys., 32 (2012), 1350-1369.  doi: 10.1017/S0143385711000241.  Google Scholar

[12]

Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, arXiv: 0804.3104v3 Google Scholar

[13]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math., 110 (1979), 529-547.  doi: 10.2307/1971237.  Google Scholar

[14]

A. Katok and B. Hasselbratt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[15]

F. Ledrappier, Propriétés ergodiques des measure de Sinai, Publ. Math. I.H.E.S., 59 (1984), 163-188.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

W. Li, J. 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.  Google Scholar

[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.  Google Scholar

[20]

V. A. Rokhlin, Lectures on the theory of entropy of transformations with invariant measures, Uspehi Mat. Nauk, 22 (1967), 3-56.   Google Scholar

[21]

D. Ruelle, Differentiation of SRB states, Commun. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.  Google Scholar

[22]

Ya. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64.   Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

J. Bochi, B. 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.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms Lecture Notes in Mathematics, 470 Springer-Verlag, New York, 1975.  Google Scholar

[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.  Google Scholar

[6]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583 Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

H. Hu, M. 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.  Google Scholar

[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.  Google Scholar

[11]

M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors, Ergod. Th. & Dynam. Sys., 32 (2012), 1350-1369.  doi: 10.1017/S0143385711000241.  Google Scholar

[12]

Y. Jiang, Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials, arXiv: 0804.3104v3 Google Scholar

[13]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math., 110 (1979), 529-547.  doi: 10.2307/1971237.  Google Scholar

[14]

A. Katok and B. Hasselbratt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[15]

F. Ledrappier, Propriétés ergodiques des measure de Sinai, Publ. Math. I.H.E.S., 59 (1984), 163-188.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

W. Li, J. 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.  Google Scholar

[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.  Google Scholar

[20]

V. A. Rokhlin, Lectures on the theory of entropy of transformations with invariant measures, Uspehi Mat. Nauk, 22 (1967), 3-56.   Google Scholar

[21]

D. Ruelle, Differentiation of SRB states, Commun. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.  Google Scholar

[22]

Ya. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64.   Google Scholar

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