Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one

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March 2015, 35(3): 967-983. doi: 10.3934/dcds.2015.35.967

Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one

Miaohua Jiang ^1,

1.	Department of Mathematics, Wake Forest University, Winston Salem, NC 27109

Received October 2013 Revised July 2014 Published October 2014

Abstract
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Under the condition that unstable manifolds are one dimensional, the derivative formula of the potential function of the generalized SRB measure with respect to the underlying dynamical system is extended from the hyperbolic attractor case to the general case when the hyperbolic set intersecting with unstable manifolds is a Cantor set. It leads to derivative formulas of objects and quantities that characterize a uniformly hyperbolic system, including the generalized SBR measure and its entropy, the root of the Bowen's equation, and the Hausdorff dimension of the hyperbolic set on a dimension two Riemannian manifold.

Keywords: SRB measure, Hausdorff dimension, potential function, linear response., entropy.

Mathematics Subject Classification: Primary: 37D20, 37D35; Secondary: 37C40, 37C4.

Citation: Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967

References:

[1]	D. Anosov and Y. Sinai, Certain smooth ergodic systems, Russian Math. Surveys, 22 (1967), 107-172.
[2]	L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in Mathematics, 294 Birkhäuser/Springer Basel AG, Basel, 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, Cambridge, 2007. doi: 10.1017/CBO9781107326026.
[4]	V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711. 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), 861-926.
[6]	D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. 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), 189-217. 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), 433-477.
[9]	M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors, Ergodic Theory Dynam. Systems, 32 (2012), 1350-1369. 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), 303-333. doi: 10.1007/s002200050814.
[11]	M. Jiang and R. de la Llave, Linear response function for coupled hyperbolic attractors, Comm. Math. Phys., 261 (2006), 379-404. doi: 10.1007/s00220-005-1446-y.
[12]	A. Katok and B. Hasselblatt, Introduction to the Modern Theorey of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[13]	R. Mañé, Erodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 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), 341-352. 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), 841-866. doi: 10.1007/BF02179463.
[16]	C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations, IMPA preprint, 2007, available at: http://w3.impa.br/~viana/out/pvw.pdf
[17]	D. Ruelle, Differentiation of SRB States, Commun. Math. Phys., 187 (1997), 227-241, Correction and complements, Comm. Math. Phys., 234 (2003), 185-190. doi: 10.1007/s002200050134.
[18]	D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics， J. Statist. Phys., 95 (1999), 393-468. 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, Cambridge, 2004.
[20]	Y. G. Sinai, Topics in Ergodic Theory, Princeton Mathematical Series, 44 Princeton University Press, Princeton, N.J., 1994.
[21]	J. Schmeling, Hölder continuity of the holonomy maps for hyperbolic basic sets. II, Math. Nachr., 170 (1994), 211-225. doi: 10.1002/mana.19941700116.

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References:

[1]	D. Anosov and Y. Sinai, Certain smooth ergodic systems, Russian Math. Surveys, 22 (1967), 107-172.
[2]	L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in Mathematics, 294 Birkhäuser/Springer Basel AG, Basel, 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, Cambridge, 2007. doi: 10.1017/CBO9781107326026.
[4]	V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711. 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), 861-926.
[6]	D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. 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), 189-217. 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), 433-477.
[9]	M. Jiang, Differentiating potential functions of SRB measures on hyperbolic attractors, Ergodic Theory Dynam. Systems, 32 (2012), 1350-1369. 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), 303-333. doi: 10.1007/s002200050814.
[11]	M. Jiang and R. de la Llave, Linear response function for coupled hyperbolic attractors, Comm. Math. Phys., 261 (2006), 379-404. doi: 10.1007/s00220-005-1446-y.
[12]	A. Katok and B. Hasselblatt, Introduction to the Modern Theorey of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[13]	R. Mañé, Erodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 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), 341-352. 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), 841-866. doi: 10.1007/BF02179463.
[16]	C. Pugh, M. Viana and A. Wilkinson, Absolute continuity of foliations, IMPA preprint, 2007, available at: http://w3.impa.br/~viana/out/pvw.pdf
[17]	D. Ruelle, Differentiation of SRB States, Commun. Math. Phys., 187 (1997), 227-241, Correction and complements, Comm. Math. Phys., 234 (2003), 185-190. doi: 10.1007/s002200050134.
[18]	D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics， J. Statist. Phys., 95 (1999), 393-468. 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, Cambridge, 2004.
[20]	Y. G. Sinai, Topics in Ergodic Theory, Princeton Mathematical Series, 44 Princeton University Press, Princeton, N.J., 1994.
[21]	J. Schmeling, Hölder continuity of the holonomy maps for hyperbolic basic sets. II, Math. Nachr., 170 (1994), 211-225. doi: 10.1002/mana.19941700116.

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