May  2014, 34(5): 2315-2332. doi: 10.3934/dcds.2014.34.2315

Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis

1. 

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

2. 

Department of Mathematics, Soochow University, Suzhou 215006

Received  November 2012 Revised  August 2013 Published  October 2013

Without constructing any measure and using properties of Markov partition, this paper provides a direct proof of dimension estimates for any subset of a limit set of a Markov construction. Furthermore, this paper investigate the dimensions of asymptotically conformal repellers. And the dimension spectrum of the level sets of nonadditive potentials on asymptotically conformal repellers are also obtained.
Citation: Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315
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.  doi: 10.1090/S0002-9947-09-04922-8.  Google Scholar

[2]

J. Barral and Y. H. Qu, Localized asymptotic behavior for almost additive potentials,, Discrete Contin. Dynam. Syst., 32 (2012), 717.  doi: 10.3934/dcds.2012.32.717.  Google Scholar

[3]

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

[4]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008).   Google Scholar

[5]

L. Barreira and P. Doutor, Dimension spectra of almost additive sequences,, Nonlinearity, 22 (2009), 2761.  doi: 10.1088/0951-7715/22/11/009.  Google Scholar

[6]

L. Barreira and P. Doutor, Almost additive multifractal analysis,, J. Math. Pures Appl., 92 (2009), 1.  doi: 10.1016/j.matpur.2009.04.006.  Google Scholar

[7]

L. Barreira and K. Gelfert, Multifractal analysis for Lyapunov exponents on nonconformal repellers,, Commun. Math. Phys., 267 (2006), 393.  doi: 10.1007/s00220-006-0084-3.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

Y. L. Cao, D. J. Feng and W. Huang, The thermodynamic formalism for submultiplicative potentials,, Discrete and Continuous Dynamical Systems, 20 (2008), 639.   Google Scholar

[12]

Y. L. Cao, The Dimension Estimate of Non-Conformal Repeller,, preprint., ().   Google Scholar

[13]

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

[14]

W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically subadditive potentials under a mistake function,, Discrete and Continuous Dynamical Systems, 32 (2012), 487.  doi: 10.3934/dcds.2012.32.487.  Google Scholar

[15]

K. Falconer, Fractal Geometry,, Mathematical foundations and applications. John Wiley & Sons, (1990).   Google Scholar

[16]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices,, Israel Journal of Mathematics, 138 (2003), 353.  doi: 10.1007/BF02783432.  Google Scholar

[17]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices,, Israel Journal of Mathematics, 170 (2009), 355.  doi: 10.1007/s11856-009-0033-x.  Google Scholar

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D. Feng and W. Huang, Lyapunov spectrum of asymptoticaliy sub-additive potentials,, Comm. Math. Phys., 297 (2010), 1.  doi: 10.1007/s00220-010-1031-x.  Google Scholar

[19]

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

[20]

K. Gelfert, Dimension estimates beyond conformal and hyperbolic dynamics,, Dyn. Syst., 20 (2005), 267.  doi: 10.1080/14689360500133142.  Google Scholar

[21]

G. Iommi, Multifractal analysis of the Lyapunov exponent for the backward continued fraction map,, Ergod. Th. Dynam. Syst., 30 (2010), 211.  doi: 10.1017/S0143385708001090.  Google Scholar

[22]

A. Käenmäki and M. Vilppolainen, Separation conditions on controlled Moran constructions,, Fund. Math., 200 (2008), 69.  doi: 10.4064/fm200-1-2.  Google Scholar

[23]

N. Luzia, A variational principle for dimension of a class of non-conformal repellers,, Ergod. Th. Dynam. Syst., 26 (2006), 821.  doi: 10.1017/S0143385705000659.  Google Scholar

[24]

P. Moran, Additive functions of intervals and Hausdorff measure,, Proc. Cambridge Philos. Soc., 42 (1946), 15.  doi: 10.1017/S0305004100022684.  Google Scholar

[25]

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture,, Comm. Math. Phys., 182 (1996), 105.  doi: 10.1007/BF02506387.  Google Scholar

[26]

Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications,, Chicago Lectures in Mathematics. University of Chicago Press, (1997).   Google Scholar

[27]

M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation,, Comm. Math. Phys., 207 (1999), 145.  doi: 10.1007/s002200050722.  Google Scholar

[28]

D. Ruelle, Repellers for real analytic maps,, Ergod. Th. Dyn. Syst., 2 (1982), 99.  doi: 10.1017/S0143385700009603.  Google Scholar

[29]

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

[30]

R. Shafikov and C. Wolf, Stable sets, hyperbolicity and dimension,, Discrete Contin. Dyn. Syst., 12 (2005), 403.   Google Scholar

[31]

P. Walters, An introduction to ergodic theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

[32]

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

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.  doi: 10.1090/S0002-9947-09-04922-8.  Google Scholar

[2]

J. Barral and Y. H. Qu, Localized asymptotic behavior for almost additive potentials,, Discrete Contin. Dynam. Syst., 32 (2012), 717.  doi: 10.3934/dcds.2012.32.717.  Google Scholar

[3]

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

[4]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008).   Google Scholar

[5]

L. Barreira and P. Doutor, Dimension spectra of almost additive sequences,, Nonlinearity, 22 (2009), 2761.  doi: 10.1088/0951-7715/22/11/009.  Google Scholar

[6]

L. Barreira and P. Doutor, Almost additive multifractal analysis,, J. Math. Pures Appl., 92 (2009), 1.  doi: 10.1016/j.matpur.2009.04.006.  Google Scholar

[7]

L. Barreira and K. Gelfert, Multifractal analysis for Lyapunov exponents on nonconformal repellers,, Commun. Math. Phys., 267 (2006), 393.  doi: 10.1007/s00220-006-0084-3.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

Y. L. Cao, D. J. Feng and W. Huang, The thermodynamic formalism for submultiplicative potentials,, Discrete and Continuous Dynamical Systems, 20 (2008), 639.   Google Scholar

[12]

Y. L. Cao, The Dimension Estimate of Non-Conformal Repeller,, preprint., ().   Google Scholar

[13]

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

[14]

W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically subadditive potentials under a mistake function,, Discrete and Continuous Dynamical Systems, 32 (2012), 487.  doi: 10.3934/dcds.2012.32.487.  Google Scholar

[15]

K. Falconer, Fractal Geometry,, Mathematical foundations and applications. John Wiley & Sons, (1990).   Google Scholar

[16]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices,, Israel Journal of Mathematics, 138 (2003), 353.  doi: 10.1007/BF02783432.  Google Scholar

[17]

D. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices,, Israel Journal of Mathematics, 170 (2009), 355.  doi: 10.1007/s11856-009-0033-x.  Google Scholar

[18]

D. Feng and W. Huang, Lyapunov spectrum of asymptoticaliy sub-additive potentials,, Comm. Math. Phys., 297 (2010), 1.  doi: 10.1007/s00220-010-1031-x.  Google Scholar

[19]

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

[20]

K. Gelfert, Dimension estimates beyond conformal and hyperbolic dynamics,, Dyn. Syst., 20 (2005), 267.  doi: 10.1080/14689360500133142.  Google Scholar

[21]

G. Iommi, Multifractal analysis of the Lyapunov exponent for the backward continued fraction map,, Ergod. Th. Dynam. Syst., 30 (2010), 211.  doi: 10.1017/S0143385708001090.  Google Scholar

[22]

A. Käenmäki and M. Vilppolainen, Separation conditions on controlled Moran constructions,, Fund. Math., 200 (2008), 69.  doi: 10.4064/fm200-1-2.  Google Scholar

[23]

N. Luzia, A variational principle for dimension of a class of non-conformal repellers,, Ergod. Th. Dynam. Syst., 26 (2006), 821.  doi: 10.1017/S0143385705000659.  Google Scholar

[24]

P. Moran, Additive functions of intervals and Hausdorff measure,, Proc. Cambridge Philos. Soc., 42 (1946), 15.  doi: 10.1017/S0305004100022684.  Google Scholar

[25]

Ya. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture,, Comm. Math. Phys., 182 (1996), 105.  doi: 10.1007/BF02506387.  Google Scholar

[26]

Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications,, Chicago Lectures in Mathematics. University of Chicago Press, (1997).   Google Scholar

[27]

M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation,, Comm. Math. Phys., 207 (1999), 145.  doi: 10.1007/s002200050722.  Google Scholar

[28]

D. Ruelle, Repellers for real analytic maps,, Ergod. Th. Dyn. Syst., 2 (1982), 99.  doi: 10.1017/S0143385700009603.  Google Scholar

[29]

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

[30]

R. Shafikov and C. Wolf, Stable sets, hyperbolicity and dimension,, Discrete Contin. Dyn. Syst., 12 (2005), 403.   Google Scholar

[31]

P. Walters, An introduction to ergodic theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

[32]

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

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