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Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one
1. | Department of Mathematics, Wake Forest University, Winston Salem, NC 27109 |
References:
[1] |
D. Anosov and Y. Sinai, Certain smooth ergodic systems,, Russian Math. Surveys, 22 (1967), 107.
|
[2] |
L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in Mathematics, 294, Birkhäuser/Springer Basel AG, (2011).
doi: 10.1007/978-3-0348-0206-2. |
[3] |
L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, (2007).
doi: 10.1017/CBO9781107326026. |
[4] |
V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps,, Nonlinearity, 21 (2008), 677.
doi: 10.1088/0951-7715/21/4/003. |
[5] |
V. Baladi and D. Smania, Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps,, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 861.
|
[6] |
D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems,, Invent. Math., 155 (2004), 389.
doi: 10.1007/s00222-003-0324-5. |
[7] |
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergodic Theory Dynam. Systems, 26 (2006), 189.
doi: 10.1017/S0143385705000374. |
[8] |
S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties,, J. Differential Geom., 79 (2008), 433.
|
[9] |
M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors,, Ergodic Theory Dynam. Systems, 32 (2012), 1350.
doi: 10.1017/S0143385711000241. |
[10] |
M. Jiang and R. de la Llave, Smooth dependence of thermodynamic limits of SRB measures,, Comm. Math. Physics, 211 (2000), 303.
doi: 10.1007/s002200050814. |
[11] |
M. Jiang and R. de la Llave, Linear response function for coupled hyperbolic attractors,, Comm. Math. Phys., 261 (2006), 379.
doi: 10.1007/s00220-005-1446-y. |
[12] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theorey of Dynamical Systems,, Cambridge University Press, (1995).
doi: 10.1017/CBO9780511809187. |
[13] |
R. Mañé, Erodic Theory and Differentiable Dynamics,, Springer-Verlag, (1987).
doi: 10.1007/978-3-642-70335-5. |
[14] |
A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics,, Bull. London Math. Soc., 34 (2002), 341.
doi: 10.1112/S0024609301008670. |
[15] |
M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets,, J. Statist. Phys., 77 (1994), 841.
doi: 10.1007/BF02179463. |
[16] |
C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations,, IMPA preprint, (2007). Google Scholar |
[17] |
D. Ruelle, Differentiation of SRB States,, Commun. Math. Phys., 187 (1997), 227.
doi: 10.1007/s002200050134. |
[18] |
D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics,, J. Statist. Phys., 95 (1999), 393.
doi: 10.1023/A:1004593915069. |
[19] |
D. Ruelle, Thermodynamic Formalism,, The mathematical structures of equilibrium statistical mechanics. Second edition. Cambridge Mathematical Library. Cambridge University Press, (2004).
|
[20] |
Y. G. Sinai, Topics in Ergodic Theory, Princeton Mathematical Series, 44, Princeton University Press, (1994).
|
[21] |
J. Schmeling, Hölder continuity of the holonomy maps for hyperbolic basic sets. II,, Math. Nachr., 170 (1994), 211.
doi: 10.1002/mana.19941700116. |
show all references
References:
[1] |
D. Anosov and Y. Sinai, Certain smooth ergodic systems,, Russian Math. Surveys, 22 (1967), 107.
|
[2] |
L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in Mathematics, 294, Birkhäuser/Springer Basel AG, (2011).
doi: 10.1007/978-3-0348-0206-2. |
[3] |
L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, (2007).
doi: 10.1017/CBO9781107326026. |
[4] |
V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps,, Nonlinearity, 21 (2008), 677.
doi: 10.1088/0951-7715/21/4/003. |
[5] |
V. Baladi and D. Smania, Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps,, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 861.
|
[6] |
D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems,, Invent. Math., 155 (2004), 389.
doi: 10.1007/s00222-003-0324-5. |
[7] |
S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergodic Theory Dynam. Systems, 26 (2006), 189.
doi: 10.1017/S0143385705000374. |
[8] |
S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties,, J. Differential Geom., 79 (2008), 433.
|
[9] |
M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors,, Ergodic Theory Dynam. Systems, 32 (2012), 1350.
doi: 10.1017/S0143385711000241. |
[10] |
M. Jiang and R. de la Llave, Smooth dependence of thermodynamic limits of SRB measures,, Comm. Math. Physics, 211 (2000), 303.
doi: 10.1007/s002200050814. |
[11] |
M. Jiang and R. de la Llave, Linear response function for coupled hyperbolic attractors,, Comm. Math. Phys., 261 (2006), 379.
doi: 10.1007/s00220-005-1446-y. |
[12] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theorey of Dynamical Systems,, Cambridge University Press, (1995).
doi: 10.1017/CBO9780511809187. |
[13] |
R. Mañé, Erodic Theory and Differentiable Dynamics,, Springer-Verlag, (1987).
doi: 10.1007/978-3-642-70335-5. |
[14] |
A. A. Pinto and D. A. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics,, Bull. London Math. Soc., 34 (2002), 341.
doi: 10.1112/S0024609301008670. |
[15] |
M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets,, J. Statist. Phys., 77 (1994), 841.
doi: 10.1007/BF02179463. |
[16] |
C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations,, IMPA preprint, (2007). Google Scholar |
[17] |
D. Ruelle, Differentiation of SRB States,, Commun. Math. Phys., 187 (1997), 227.
doi: 10.1007/s002200050134. |
[18] |
D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics,, J. Statist. Phys., 95 (1999), 393.
doi: 10.1023/A:1004593915069. |
[19] |
D. Ruelle, Thermodynamic Formalism,, The mathematical structures of equilibrium statistical mechanics. Second edition. Cambridge Mathematical Library. Cambridge University Press, (2004).
|
[20] |
Y. G. Sinai, Topics in Ergodic Theory, Princeton Mathematical Series, 44, Princeton University Press, (1994).
|
[21] |
J. Schmeling, Hölder continuity of the holonomy maps for hyperbolic basic sets. II,, Math. Nachr., 170 (1994), 211.
doi: 10.1002/mana.19941700116. |
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