# American Institute of Mathematical Sciences

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

## Infimum of the metric entropy of volume preserving Anosov systems

 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.

Citation: Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete & Continuous Dynamical Systems - A, 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. [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. 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. [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. 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. [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. 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. [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.

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##### 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. [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. 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. [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. 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. [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. 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. [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.
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