February  2020, 40(2): 883-905. doi: 10.3934/dcds.2020065

Dimensions of $ C^1- $average conformal hyperbolic sets

1. 

School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China

2. 

Departament of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, China

3. 

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

4. 

Center for Dynamical Systems and Differential Equation, Soochow University, Suzhou 215006, Jiangsu, China

* Corresponding author: Yun Zhao

Received  March 2019 Revised  August 2019 Published  November 2019

Fund Project: The first author is supported by NSFC (11501400, 11871361) and the Talent Program of Shanghai University of Engineering Science. The third author is partially supported by NSFC (11771317, 11790274) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000). The fourth author is partially supported by NSFC (11871361, 11790274).

This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a $ C^1 $ diffeomorphism, utilizing the techniques in sub-additive thermodynamic formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which is exactly the sum of the dimensions of the restriction of the hyperbolic set to stable and unstable manifolds. Furthermore, the dimensions of an average conformal hyperbolic set vary continuously with respect to the dynamics.

Citation: Juan Wang, Jing Wang, Yongluo Cao, Yun Zhao. Dimensions of $ C^1- $average conformal hyperbolic sets. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 883-905. doi: 10.3934/dcds.2020065
References:
[1]

J. BanY. Cao and H. Hu, The dimension 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, Ergodic Theory Dynam. Systems, 16 (1996), 871-927.  doi: 10.1017/S0143385700010117.

[3]

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

[4]

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.

[5]

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

[6]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.

[7]

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

[8]

Y. CaoD. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657.  doi: 10.3934/dcds.2008.20.639.

[9]

Y. CaoY. Pesin and Y. Zhao, Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure, Geom. Funct. Anal., 29 (2019), 1325-1368.  doi: 10.1007/s00039-019-00510-7.

[10]

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.

[11]

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

[12]

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

[13]

K. Falconer, Bounded distortion and dimension for nonconformal repellers, Math. Proc. Cambridge Philos. Soc., 115 (1994), 315-334.  doi: 10.1017/S030500410007211X.

[14]

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

[15]

W. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. symp. pure Math., 14 (1970), 133-163. 

[16]

W. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, Berlin-New York, 1977.

[17] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.
[18]

H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems, 3 (1983), 251-260.  doi: 10.1017/S0143385700001966.

[19]

V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 197-231. 

[20]

J. Palis and M. Viana, On the continuity of hausdorff dimension and limit capacity for horseshoes, in Dynamical Systems, Valparaiso 1986, Lecture Notes in Mathematics, Springer, Berlin, 1331 (1988), 150–160. doi: 10.1007/BFb0083071.

[21] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.
[22] F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9781139193184.
[23]

C. PughM. Shub and A. Wilkinson, Hölder foliations, Duke Mathematical Journal, 86 (1997), 517-546.  doi: 10.1215/S0012-7094-97-08616-6.

[24]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2 (1982), 99-107.  doi: 10.1017/S0143385700009603.

[25]

F. Takens, Limit capacity and hausdorff dimension of dynamically defined cantor sets, in Dynamical systems, Valparaiso 1986, Lecture Notes in Mathematics, Springer, Berlin, 1331 (1988), 196–212. doi: 10.1007/BFb0083074.

[26]

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

[27]

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

[28]

Y. ZhaoY. Cao and J. Ban, The Hausdorff dimension of average conformal repellers under random perturbation, Nonlinearity, 22 (2009), 2405-2416.  doi: 10.1088/0951-7715/22/10/005.

show all references

References:
[1]

J. BanY. Cao and H. Hu, The dimension 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, Ergodic Theory Dynam. Systems, 16 (1996), 871-927.  doi: 10.1017/S0143385700010117.

[3]

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

[4]

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.

[5]

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

[6]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.

[7]

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

[8]

Y. CaoD. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst., 20 (2008), 639-657.  doi: 10.3934/dcds.2008.20.639.

[9]

Y. CaoY. Pesin and Y. Zhao, Dimension estimates for non-conformal repellers and continuity of sub-additive topological pressure, Geom. Funct. Anal., 29 (2019), 1325-1368.  doi: 10.1007/s00039-019-00510-7.

[10]

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.

[11]

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

[12]

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

[13]

K. Falconer, Bounded distortion and dimension for nonconformal repellers, Math. Proc. Cambridge Philos. Soc., 115 (1994), 315-334.  doi: 10.1017/S030500410007211X.

[14]

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

[15]

W. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. symp. pure Math., 14 (1970), 133-163. 

[16]

W. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, Berlin-New York, 1977.

[17] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.
[18]

H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems, 3 (1983), 251-260.  doi: 10.1017/S0143385700001966.

[19]

V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trans. Mosc. Math. Soc., 19 (1968), 197-231. 

[20]

J. Palis and M. Viana, On the continuity of hausdorff dimension and limit capacity for horseshoes, in Dynamical Systems, Valparaiso 1986, Lecture Notes in Mathematics, Springer, Berlin, 1331 (1988), 150–160. doi: 10.1007/BFb0083071.

[21] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.
[22] F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9781139193184.
[23]

C. PughM. Shub and A. Wilkinson, Hölder foliations, Duke Mathematical Journal, 86 (1997), 517-546.  doi: 10.1215/S0012-7094-97-08616-6.

[24]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems, 2 (1982), 99-107.  doi: 10.1017/S0143385700009603.

[25]

F. Takens, Limit capacity and hausdorff dimension of dynamically defined cantor sets, in Dynamical systems, Valparaiso 1986, Lecture Notes in Mathematics, Springer, Berlin, 1331 (1988), 196–212. doi: 10.1007/BFb0083074.

[26]

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

[27]

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

[28]

Y. ZhaoY. Cao and J. Ban, The Hausdorff dimension of average conformal repellers under random perturbation, Nonlinearity, 22 (2009), 2405-2416.  doi: 10.1088/0951-7715/22/10/005.

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