American Institute of Mathematical Sciences

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

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

Juan Wang 1, , Jing Wang 3,4, , Yongluo Cao 2,3,4, and  Yun Zhao 3,4,,

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.

Keywords: Dimension theory, average conformal, hyperbolic set, topological pressure.
Mathematics Subject Classification: Primary: 37C45, 37D20; Secondary: 37D35.
Citation: Juan Wang, Jing Wang, Yongluo Cao, Yun Zhao. Dimensions of $ C^1- $average conformal hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 883-905. doi: 10.3934/dcds.2020065
References:
[1]

J. Ban, Y. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

Y. Cao, D. 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.  Google Scholar

[9]

Y. Cao, Y. 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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar
[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar
[22] F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9781139193184.  Google Scholar
[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

Y. Zhao, Y. 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.  Google Scholar

show all references

References:
[1]

J. Ban, Y. 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

Y. Cao, D. 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.  Google Scholar

[9]

Y. Cao, Y. 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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar
[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar
[22] F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9781139193184.  Google Scholar
[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

Y. Zhao, Y. 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.  Google Scholar

[1]

Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[2]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[3]

Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in non-conformal setting. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3847-3873. doi: 10.3934/dcds.2014.34.3847
[4]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
[5]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure & Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805
[6]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[7]

Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250
[8]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[9]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[10]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[11]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[12]

Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058
[13]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[14]

Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201
[15]

Boris Hasselblatt and Jorg Schmeling. Dimension product structure of hyperbolic sets. Electronic Research Announcements, 2004, 10: 88-96.

[16]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[17]

Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic & Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004
[18]

Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365
[19]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309
[20]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (27)
  • HTML views (39)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences