# American Institute of Mathematical Sciences

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. [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. [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. [4] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008). [5] L. Barreira and P. Doutor, Dimension spectra of almost additive sequences,, Nonlinearity, 22 (2009), 2761. doi: 10.1088/0951-7715/22/11/009. [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. [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. [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. [9] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. [10] R. Bowen, Hausdorff dimension of quasicircles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11. [11] Y. L. Cao, D. J. Feng and W. Huang, The thermodynamic formalism for submultiplicative potentials,, Discrete and Continuous Dynamical Systems, 20 (2008), 639. [12] Y. L. Cao, The Dimension Estimate of Non-Conformal Repeller,, preprint., (). [13] J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey,, Nonlinearity, 23 (2010). doi: 10.1088/0951-7715/23/4/R01. [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. [15] K. Falconer, Fractal Geometry,, Mathematical foundations and applications. John Wiley & Sons, (1990). [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. [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. [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. [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. [20] K. Gelfert, Dimension estimates beyond conformal and hyperbolic dynamics,, Dyn. Syst., 20 (2005), 267. doi: 10.1080/14689360500133142. [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. [22] A. Käenmäki and M. Vilppolainen, Separation conditions on controlled Moran constructions,, Fund. Math., 200 (2008), 69. doi: 10.4064/fm200-1-2. [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. [24] P. Moran, Additive functions of intervals and Hausdorff measure,, Proc. Cambridge Philos. Soc., 42 (1946), 15. doi: 10.1017/S0305004100022684. [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. [26] Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications,, Chicago Lectures in Mathematics. University of Chicago Press, (1997). [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. [28] D. Ruelle, Repellers for real analytic maps,, Ergod. Th. Dyn. Syst., 2 (1982), 99. doi: 10.1017/S0143385700009603. [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. [30] R. Shafikov and C. Wolf, Stable sets, hyperbolicity and dimension,, Discrete Contin. Dyn. Syst., 12 (2005), 403. [31] P. Walters, An introduction to ergodic theory,, Graduate Texts in Mathematics, (1982). [32] Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets,, Ergod. Th. Dynam. Syst., 17 (1997), 739. doi: 10.1017/S0143385797085003.

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##### 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. [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. [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. [4] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008). [5] L. Barreira and P. Doutor, Dimension spectra of almost additive sequences,, Nonlinearity, 22 (2009), 2761. doi: 10.1088/0951-7715/22/11/009. [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. [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. [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. [9] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. [10] R. Bowen, Hausdorff dimension of quasicircles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11. [11] Y. L. Cao, D. J. Feng and W. Huang, The thermodynamic formalism for submultiplicative potentials,, Discrete and Continuous Dynamical Systems, 20 (2008), 639. [12] Y. L. Cao, The Dimension Estimate of Non-Conformal Repeller,, preprint., (). [13] J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey,, Nonlinearity, 23 (2010). doi: 10.1088/0951-7715/23/4/R01. [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. [15] K. Falconer, Fractal Geometry,, Mathematical foundations and applications. John Wiley & Sons, (1990). [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. [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. [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. [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. [20] K. Gelfert, Dimension estimates beyond conformal and hyperbolic dynamics,, Dyn. Syst., 20 (2005), 267. doi: 10.1080/14689360500133142. [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. [22] A. Käenmäki and M. Vilppolainen, Separation conditions on controlled Moran constructions,, Fund. Math., 200 (2008), 69. doi: 10.4064/fm200-1-2. [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. [24] P. Moran, Additive functions of intervals and Hausdorff measure,, Proc. Cambridge Philos. Soc., 42 (1946), 15. doi: 10.1017/S0305004100022684. [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. [26] Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications,, Chicago Lectures in Mathematics. University of Chicago Press, (1997). [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. [28] D. Ruelle, Repellers for real analytic maps,, Ergod. Th. Dyn. Syst., 2 (1982), 99. doi: 10.1017/S0143385700009603. [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. [30] R. Shafikov and C. Wolf, Stable sets, hyperbolicity and dimension,, Discrete Contin. Dyn. Syst., 12 (2005), 403. [31] P. Walters, An introduction to ergodic theory,, Graduate Texts in Mathematics, (1982). [32] Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets,, Ergod. Th. Dynam. Syst., 17 (1997), 739. doi: 10.1017/S0143385797085003.
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