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Entropy of diffeomorphisms of line
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 |
In this paper we continue our study [
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.
|
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. |
[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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