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September  2014, 34(9): 3847-3873. doi: 10.3934/dcds.2014.34.3847

Dimension estimates in non-conformal setting

1. 

Department of mathematics, Soochow University, Suzhou 215006, Jiangsu, China

2. 

Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China

Received  July 2013 Revised  December 2013 Published  March 2014

Given an arbitrary subset of a non-conformal repeller of a $C^1$ map. Using directly the definitions of Hausdorff dimension and Box dimension and of pressure, this paper first proves that the zeros of the topological pressure on this set give its dimension estimates. And without using the estimate of pointwise dimension of an ergodic measure on a non-conformal repeller, it is showed that the zeros of non-additive measure-theoretic pressure give the lower and upper bound of dimension estimate for it. Some results from [22,41] are extended for $C^1$ maps or arbitrary subsets of a non-conformal repeller. And a remark on Rugh's result [34] is also given in this paper.
Citation: Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in non-conformal setting. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3847-3873. doi: 10.3934/dcds.2014.34.3847
References:
[1]

J. Ban, Y. Cao and H. Hu, The dimensions of a non-conformal repeller and an average conformal repeller, Trans. Amer. Math. Soc., 362 (2010), 727-751. doi: 10.1090/S0002-9947-09-04922-8.

[2]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Syst., 16 (1996), 871-927. doi: 10.1017/S0143385700010117.

[3]

L. Barreira, Dimension estimates in nonconformal hyperbolic dynamics, Nonlinearity, 16 (2003), 1657-1672. doi: 10.1088/0951-7715/16/5/307.

[4]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, 272, Birkhäuser Verlag, Basel, 2008.

[5]

L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in Mathematics, 294, Birkhäuser, Springer Basel, 2011. doi: 10.1007/978-3-0348-0206-2.

[6]

L. Barreira and K. Gelfert, Dimension estimates in smooth dynamics: A survey of recent results, Ergod. Th. Dynam. Syst., 31 (2011), 641-671. doi: 10.1017/S014338571000012X.

[7]

L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergod. Th. Dynam. Syst., 26 (2006), 653-671. doi: 10.1017/S0143385705000672.

[8]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.

[9]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25.

[10]

R. Bowen, Equlibrium States and the Ergodic Theory of Anosov Diffeomorphism, Lecture Notes in Mathematics, 470, Springer, New York-Heidelberg-Berlin, 1975.

[11]

M. Brin and A. Katok, On Local Entropy, in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, (1983), 30-38. doi: 10.1007/BFb0061408.

[12]

Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discret. Contin. Dynam. Syst., 20 (2008), 639-657.

[13]

Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Ergod. Th. Dynam. Syst., 33 (2013), 831-850. doi: 10.1017/S0143385712000090.

[14]

Y. Cao, Dimension spectrum of asymptotically additive potentials for $C^1$ average conformal repellers, Nonlinearity, 26 (2013), 2441-2468. doi: 10.1088/0951-7715/26/9/2441.

[15]

J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey, Nonlinearity, 23 (2010), R93-R114. doi: 10.1088/0951-7715/23/4/R01.

[16]

W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically subadditive potentials under a mistake funciton, Discret. Contin. Dyn. Syst., 32 (2012), 487-497. doi: 10.3934/dcds.2012.32.487.

[17]

V. Climenhaga, Bowen's equation in the non-uniform setting, Ergod. Th. Dynam. Syst., 31 (2011), 1163-1182. doi: 10.1017/S0143385710000362.

[18]

M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4 (1991), 103-134. doi: 10.1088/0951-7715/4/1/008.

[19]

K. Falconer, Fractal Geometry-Mathematical Foundations and Applications, Second edition. John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.

[20]

K. Falconer, Bounded distortion and dimension for non-conformal repellers, Math. Proc. Camb. Phil. Soc., 115 (1994), 315-334. doi: 10.1017/S030500410007211X.

[21]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. Dynam. Syst., 17 (1997), 147-167. doi: 10.1017/S0143385797060987.

[22]

H. Hu, Dimensions of invariant sets of expanding maps, Commun. Math. Phys., 176 (1996), 307-320. doi: 10.1007/BF02099551.

[23]

W. Huang and P. Zhang, Pointwise dimension, entropy and Lyapunov exponents for $C^1$ map, Trans. Amer. Math. Soc., 364 (2012), 6355-6370. doi: 10.1090/S0002-9947-2012-05527-9.

[24]

F. Ledrappier, Some relations between dimension and Lyapunov exponent, Commun. Math. Phys., 81 (1981), 229-238. doi: 10.1007/BF01208896.

[25]

V. Mayer and M. Urbański, Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order, Ergod. Th. Dynam. Syst., 28 (2008), 915-946. doi: 10.1017/S0143385707000648.

[26]

V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Mem. Amer. Math. Soc., 203 (2010), vi+107 pp. doi: 10.1090/S0065-9266-09-00577-8.

[27]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Lyapnov exponents of dynamical systems, Trudy Moskov. Mat. Obšž, 19 (1968), 179-210.

[28]

Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago: University of Chicago Press, 1997.

[29]

F. Przytycki, J. Letelier and S. Smirnov, Equivalence and topo- logical invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63. doi: 10.1007/s00222-002-0243-x.

[30]

F. Przytycki, J. Letelier and S. Smirnov, Equality of pressures for rational functions, Ergod. Th. Dynam. Syst., 24 (2004), 891-914. doi: 10.1017/S0143385703000385.

[31]

F. Przytycki and M. Urbažski, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society, Lecture Note Series 371, Cambridge University Press, Cambridge, 2010.

[32]

D. Ruelle, Repellers for real analytic maps, Ergod. Th. Dynam. Syst., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.

[33]

D. Ruelle, An inequality for the entropy of differential maps, Bol. Soc. Bras. De Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[34]

H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency, Annals of Mathematics, 168 (2008), 695-748. doi: 10.4007/annals.2008.168.695.

[35]

M. Urbažski, On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point, Studia Math., 97 (1991), 167-188.

[36]

M. Urbažski, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.

[37]

M. Urbažski and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergod. Th. Dynam. Syst., 24 (2004), 279-315. doi: 10.1017/S0143385703000208.

[38]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.

[39]

C. Wolf, On the box dimension of an invariant set, Nonlinearity, 14 (2001), 73-79. doi: 10.1088/0951-7715/14/1/303.

[40]

L. S. Young, Dimension, entropy and Lyapunov exponents, Ergod. th. Dynam. Syst., 2 (1982), 109-124. doi: 10.1017/S0143385700009615.

[41]

Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets, Ergod. Th. Dynam. Syst., 17 (1997), 739-756. doi: 10.1017/S0143385797085003.

[42]

Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear analysis, 70 (2009), 2237-2247. doi: 10.1016/j.na.2008.03.003.

[43]

Y. Zhao, A note on the measure-theoretic pressure in subadditive case, Chinese Annals of Math., Series A, 29 (2008), 325-332.

show all references

References:
[1]

J. Ban, Y. Cao and H. Hu, The dimensions of a non-conformal repeller and an average conformal repeller, Trans. Amer. Math. Soc., 362 (2010), 727-751. doi: 10.1090/S0002-9947-09-04922-8.

[2]

L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. Dynam. Syst., 16 (1996), 871-927. doi: 10.1017/S0143385700010117.

[3]

L. Barreira, Dimension estimates in nonconformal hyperbolic dynamics, Nonlinearity, 16 (2003), 1657-1672. doi: 10.1088/0951-7715/16/5/307.

[4]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, 272, Birkhäuser Verlag, Basel, 2008.

[5]

L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory, Progress in Mathematics, 294, Birkhäuser, Springer Basel, 2011. doi: 10.1007/978-3-0348-0206-2.

[6]

L. Barreira and K. Gelfert, Dimension estimates in smooth dynamics: A survey of recent results, Ergod. Th. Dynam. Syst., 31 (2011), 641-671. doi: 10.1017/S014338571000012X.

[7]

L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions, Ergod. Th. Dynam. Syst., 26 (2006), 653-671. doi: 10.1017/S0143385705000672.

[8]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.

[9]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25.

[10]

R. Bowen, Equlibrium States and the Ergodic Theory of Anosov Diffeomorphism, Lecture Notes in Mathematics, 470, Springer, New York-Heidelberg-Berlin, 1975.

[11]

M. Brin and A. Katok, On Local Entropy, in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, (1983), 30-38. doi: 10.1007/BFb0061408.

[12]

Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discret. Contin. Dynam. Syst., 20 (2008), 639-657.

[13]

Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure, Ergod. Th. Dynam. Syst., 33 (2013), 831-850. doi: 10.1017/S0143385712000090.

[14]

Y. Cao, Dimension spectrum of asymptotically additive potentials for $C^1$ average conformal repellers, Nonlinearity, 26 (2013), 2441-2468. doi: 10.1088/0951-7715/26/9/2441.

[15]

J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey, Nonlinearity, 23 (2010), R93-R114. doi: 10.1088/0951-7715/23/4/R01.

[16]

W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically subadditive potentials under a mistake funciton, Discret. Contin. Dyn. Syst., 32 (2012), 487-497. doi: 10.3934/dcds.2012.32.487.

[17]

V. Climenhaga, Bowen's equation in the non-uniform setting, Ergod. Th. Dynam. Syst., 31 (2011), 1163-1182. doi: 10.1017/S0143385710000362.

[18]

M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4 (1991), 103-134. doi: 10.1088/0951-7715/4/1/008.

[19]

K. Falconer, Fractal Geometry-Mathematical Foundations and Applications, Second edition. John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.

[20]

K. Falconer, Bounded distortion and dimension for non-conformal repellers, Math. Proc. Camb. Phil. Soc., 115 (1994), 315-334. doi: 10.1017/S030500410007211X.

[21]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergod. Th. Dynam. Syst., 17 (1997), 147-167. doi: 10.1017/S0143385797060987.

[22]

H. Hu, Dimensions of invariant sets of expanding maps, Commun. Math. Phys., 176 (1996), 307-320. doi: 10.1007/BF02099551.

[23]

W. Huang and P. Zhang, Pointwise dimension, entropy and Lyapunov exponents for $C^1$ map, Trans. Amer. Math. Soc., 364 (2012), 6355-6370. doi: 10.1090/S0002-9947-2012-05527-9.

[24]

F. Ledrappier, Some relations between dimension and Lyapunov exponent, Commun. Math. Phys., 81 (1981), 229-238. doi: 10.1007/BF01208896.

[25]

V. Mayer and M. Urbański, Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order, Ergod. Th. Dynam. Syst., 28 (2008), 915-946. doi: 10.1017/S0143385707000648.

[26]

V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Mem. Amer. Math. Soc., 203 (2010), vi+107 pp. doi: 10.1090/S0065-9266-09-00577-8.

[27]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Lyapnov exponents of dynamical systems, Trudy Moskov. Mat. Obšž, 19 (1968), 179-210.

[28]

Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago: University of Chicago Press, 1997.

[29]

F. Przytycki, J. Letelier and S. Smirnov, Equivalence and topo- logical invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63. doi: 10.1007/s00222-002-0243-x.

[30]

F. Przytycki, J. Letelier and S. Smirnov, Equality of pressures for rational functions, Ergod. Th. Dynam. Syst., 24 (2004), 891-914. doi: 10.1017/S0143385703000385.

[31]

F. Przytycki and M. Urbažski, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society, Lecture Note Series 371, Cambridge University Press, Cambridge, 2010.

[32]

D. Ruelle, Repellers for real analytic maps, Ergod. Th. Dynam. Syst., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.

[33]

D. Ruelle, An inequality for the entropy of differential maps, Bol. Soc. Bras. De Mat., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[34]

H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency, Annals of Mathematics, 168 (2008), 695-748. doi: 10.4007/annals.2008.168.695.

[35]

M. Urbažski, On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point, Studia Math., 97 (1991), 167-188.

[36]

M. Urbažski, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.

[37]

M. Urbažski and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergod. Th. Dynam. Syst., 24 (2004), 279-315. doi: 10.1017/S0143385703000208.

[38]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.

[39]

C. Wolf, On the box dimension of an invariant set, Nonlinearity, 14 (2001), 73-79. doi: 10.1088/0951-7715/14/1/303.

[40]

L. S. Young, Dimension, entropy and Lyapunov exponents, Ergod. th. Dynam. Syst., 2 (1982), 109-124. doi: 10.1017/S0143385700009615.

[41]

Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets, Ergod. Th. Dynam. Syst., 17 (1997), 739-756. doi: 10.1017/S0143385797085003.

[42]

Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials, Nonlinear analysis, 70 (2009), 2237-2247. doi: 10.1016/j.na.2008.03.003.

[43]

Y. Zhao, A note on the measure-theoretic pressure in subadditive case, Chinese Annals of Math., Series A, 29 (2008), 325-332.

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