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

1. 

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

Received  October 2013 Revised  July 2014 Published  October 2014

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.
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.

show all references

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.

[1]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[2]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

[3]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[4]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060

[5]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

[6]

Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647

[7]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018

[8]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[9]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591

[10]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503

[11]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405

[12]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[13]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293

[14]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191

[15]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192

[16]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012

[17]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098

[18]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125

[19]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015

[20]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (102)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]