October  2017, 37(10): 5367-5405. doi: 10.3934/dcds.2017234

Multifractal analysis of random weak Gibbs measures

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received  August 2016 Revised  May 2017 Published  June 2017

We describe the multifractal nature of random weak Gibbs measures on some classes of attractors associated with $C^1$ random dynamics semi-conjugate to a random subshift of finite type. This includes the validity of the multifractal formalism, the calculation of Hausdorff and packing dimensions of the so-called level sets of divergent points, and a $0$-$∞$ law for the Hausdorff and packing measures of the level sets of the local dimension.

Citation: Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234
References:
[1]

I. S. BaekL. Olsen and N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math., 214 (2007), 267-287.  doi: 10.1016/j.aim.2007.02.003.  Google Scholar

[2]

J. Barral, Inverse problems in multifractal analysis, Ann. Sci. Ec. Norm. Sup.(4), 48 (2015), 1457-1510.  doi: 10.24033/asens.2274.  Google Scholar

[3]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam, 1 (1992/93), 99-116.   Google Scholar

[4]

T. Bogenschütz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type, Ergod. Th. & Dynam. Sys., 15 (1995), 413-447.  doi: 10.1017/S0143385700008464.  Google Scholar

[5]

P. ColletJ. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys., 47 (1987), 609-644.  doi: 10.1007/BF01206149.  Google Scholar

[6]

M. Denker and M. Gordin, Gibbs measures for fibred systems, Adv. Math., 148 (1999), 161-192.  doi: 10.1006/aima.1999.1843.  Google Scholar

[7]

M. DenkerY. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 22 (2008), 131-164.  doi: 10.3934/dcds.2008.22.131.  Google Scholar

[8]

M. DenkerY. Kifer and M. Stadlbauer, Corrigendum to: Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 35 (2015), 593-594.  doi: 10.3934/dcds.2015.35.593.  Google Scholar

[9]

K. J. Falconer, Techniques in Fractal Geometry John Wiley & Sons, Ltd., Chichester, 1997.  Google Scholar

[10]

A.-H. FanD.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[11]

D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part Ⅰ: Positive matrices, Israel J. Math., 138 (2003), 353-376.  doi: 10.1007/BF02783432.  Google Scholar

[12]

D.-J. Feng, Multifractal analysis of Bernoulli convolutions associated with Salem numbers, Adv. Math., 229 (2012), 3052-3077.  doi: 10.1016/j.aim.2011.11.006.  Google Scholar

[13]

D.-J. FengK.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.  doi: 10.1006/aima.2001.2054.  Google Scholar

[14]

D.-J. Feng and E. Olivier, Multifractal analysis of weak Gibbs measures and phase transition– application to some Bernoulli convolutions, Ergod. Th. & Dynam. Sys., 23 (2003), 1751-1784.  doi: 10.1017/S0143385703000051.  Google Scholar

[15]

D.-J. Feng and L. Shu, Multifractal analysis for disintegrations of Gibbs measures and conditional Birkhoff averages, Ergod. Th. & Dynam. Sys., 29 (2009), 885-918.  doi: 10.1017/S0143385708000655.  Google Scholar

[16]

D.-J. Feng and J. Wu, The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity, 14 (2001), 81-85.  doi: 10.1088/0951-7715/14/1/304.  Google Scholar

[17]

U. Frisch and G. Parisi, Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics, International school of Physics, 41 (1985), 84-88.   Google Scholar

[18]

V. M. Gundlach, Thermodynamic Formalism for random subshift of finite type, 1996. Google Scholar

[19]

T. C. HalseyM. H. JensenL. P. KadanoffI. Procaccia and B. I. Shraiman, Fractal measures and their singularities: The characterisation of strange sets, Phys. Rev. A, 33 (1986), 1141-1151.  doi: 10.1103/PhysRevA.33.1141.  Google Scholar

[20]

M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra, Nonlinearity, 14 (2001), 395-409.  doi: 10.1088/0951-7715/14/2/312.  Google Scholar

[21]

K. M. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2,171, Amer. Math. Soc., Providence, RI (1996), 107–140. doi: 10.1090/trans2/171/10.  Google Scholar

[22]

Y. Kifer, Equilibrium states for random expanding transformations, Random Comput. Dynam., 1 (1992/93), 1-31.   Google Scholar

[23]

Y. Kifer, Fractals via random iterated function systems and random geometric constructions, in Fractal geometry and stochastics (Finsterbergen, 1994), Progr. Probab., Birkhäuser, Basel, 37 (1995), 145–164. doi: 10.1007/978-3-0348-7755-8_7.  Google Scholar

[24]

Y. Kifer, Fractal dimensions and random transformations, Trans. Amer. Math. Soc., 348 (1996), 2003-2038.  doi: 10.1090/S0002-9947-96-01608-X.  Google Scholar

[25]

Y. Kifer, On the topological pressure for random bundle transformations, in Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 202 (2001), 197–214. doi: 10.1090/trans2/202/14.  Google Scholar

[26]

Y. Kifer, Thermodynamic formalism for random transformations revisited, Stoch. Dyn., 8 (2008), 77-102.  doi: 10.1142/S0219493708002238.  Google Scholar

[27]

Y. Kifer and P. -D. Liu, Random dynamics, in Handbook of dynamical systems, Elsevier B. V., Amsterdam, 1B (2006), 379–499. doi: 10.1016/S1874-575X(06)80030-5.  Google Scholar

[28]

S. P. Lalley and D. Gatzouras, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J., 41 (1992), 533-568.  doi: 10.1512/iumj.1992.41.41031.  Google Scholar

[29]

K.-S. Lau and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math., 141 (1999), 45-96.  doi: 10.1006/aima.1998.1773.  Google Scholar

[30]

N. Luzia, A variational principle for the dimension for a class of non-conformal repellers, Ergod. Th. & Dynam. Sys., 26 (2006), 821-845.  doi: 10.1017/S0143385705000659.  Google Scholar

[31]

J.-H. MaZ.-Y. Wen and J. Wu, Besicovitch subsets of self-similar sets, Ann. Inst. Fourier, 52 (2002), 1061-1074.  doi: 10.5802/aif.1911.  Google Scholar

[32]

J.-H. Ma and Z.-Y. Wen, Hausdorff and Packing measure of sets of generic points: A ZeroInfinity Law, J. London Math. Soc., 69 (2004), 383-406.  doi: 10.1112/S0024610703005040.  Google Scholar

[33]

P. T. Maker, The ergodic theorem for a sequence of functions, Duke Math. J., 6 (1940), 27-30.  doi: 10.1215/S0012-7094-40-00602-0.  Google Scholar

[34]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[35]

V. Mayer, B. Skorulski and M. Urbański, Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry Lecture Notes in Mathematics, 2036, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-23650-1.  Google Scholar

[36]

E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures, Nonlinearity, 12 (1999), 1571-1585.  doi: 10.1088/0951-7715/12/6/309.  Google Scholar

[37]

L. Olsen, A multifractal formalism, Adv. Math., 116 (1995), 82-196.  doi: 10.1006/aima.1995.1066.  Google Scholar

[38]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.  doi: 10.1016/j.matpur.2003.09.007.  Google Scholar

[39]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. Ⅳ: Divergence points and packing dimension, Bull. Sci. Math., 132 (2008), 650-678.  doi: 10.1016/j.bulsci.2008.08.002.  Google Scholar

[40]

N. Patzschke, Self-Conformal Multifractal Measures, Adv. Appl. Math., 19 (1997), 486-513.  doi: 10.1006/aama.1997.0557.  Google Scholar

[41]

Y. 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), 105-153.  doi: 10.1007/BF02506387.  Google Scholar

[42]

Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys., 86 (1997), 233-275.  doi: 10.1007/BF02180206.  Google Scholar

[43]

D. A. Rand, The singularity spectrum f(α) for cookie-cutters, Ergod. Th. & Dynam. Sys., 9 (1989), 527–541. doi: 10.1017/S0143385700005162.  Google Scholar

[44]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Th. & Dynam. Sys., 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820.  Google Scholar

[45]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Hyperbolic dynamics, fluctuations and large deviations, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 89 (2015), 81-117.  doi: 10.1090/pspum/089/01485.  Google Scholar

[46]

L. Shu, The multifractal analysis of Birkhoff averages for conformal repellers under random perturbations, Monatsh. Math., 159 (2010), 81-113.  doi: 10.1007/s00605-009-0149-4.  Google Scholar

show all references

References:
[1]

I. S. BaekL. Olsen and N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math., 214 (2007), 267-287.  doi: 10.1016/j.aim.2007.02.003.  Google Scholar

[2]

J. Barral, Inverse problems in multifractal analysis, Ann. Sci. Ec. Norm. Sup.(4), 48 (2015), 1457-1510.  doi: 10.24033/asens.2274.  Google Scholar

[3]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam, 1 (1992/93), 99-116.   Google Scholar

[4]

T. Bogenschütz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type, Ergod. Th. & Dynam. Sys., 15 (1995), 413-447.  doi: 10.1017/S0143385700008464.  Google Scholar

[5]

P. ColletJ. L. Lebowitz and A. Porzio, The dimension spectrum of some dynamical systems, J. Statist. Phys., 47 (1987), 609-644.  doi: 10.1007/BF01206149.  Google Scholar

[6]

M. Denker and M. Gordin, Gibbs measures for fibred systems, Adv. Math., 148 (1999), 161-192.  doi: 10.1006/aima.1999.1843.  Google Scholar

[7]

M. DenkerY. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 22 (2008), 131-164.  doi: 10.3934/dcds.2008.22.131.  Google Scholar

[8]

M. DenkerY. Kifer and M. Stadlbauer, Corrigendum to: Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 35 (2015), 593-594.  doi: 10.3934/dcds.2015.35.593.  Google Scholar

[9]

K. J. Falconer, Techniques in Fractal Geometry John Wiley & Sons, Ltd., Chichester, 1997.  Google Scholar

[10]

A.-H. FanD.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[11]

D.-J. Feng, Lyapunov exponents for products of matrices and multifractal analysis. Part Ⅰ: Positive matrices, Israel J. Math., 138 (2003), 353-376.  doi: 10.1007/BF02783432.  Google Scholar

[12]

D.-J. Feng, Multifractal analysis of Bernoulli convolutions associated with Salem numbers, Adv. Math., 229 (2012), 3052-3077.  doi: 10.1016/j.aim.2011.11.006.  Google Scholar

[13]

D.-J. FengK.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.  doi: 10.1006/aima.2001.2054.  Google Scholar

[14]

D.-J. Feng and E. Olivier, Multifractal analysis of weak Gibbs measures and phase transition– application to some Bernoulli convolutions, Ergod. Th. & Dynam. Sys., 23 (2003), 1751-1784.  doi: 10.1017/S0143385703000051.  Google Scholar

[15]

D.-J. Feng and L. Shu, Multifractal analysis for disintegrations of Gibbs measures and conditional Birkhoff averages, Ergod. Th. & Dynam. Sys., 29 (2009), 885-918.  doi: 10.1017/S0143385708000655.  Google Scholar

[16]

D.-J. Feng and J. Wu, The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity, 14 (2001), 81-85.  doi: 10.1088/0951-7715/14/1/304.  Google Scholar

[17]

U. Frisch and G. Parisi, Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics, International school of Physics, 41 (1985), 84-88.   Google Scholar

[18]

V. M. Gundlach, Thermodynamic Formalism for random subshift of finite type, 1996. Google Scholar

[19]

T. C. HalseyM. H. JensenL. P. KadanoffI. Procaccia and B. I. Shraiman, Fractal measures and their singularities: The characterisation of strange sets, Phys. Rev. A, 33 (1986), 1141-1151.  doi: 10.1103/PhysRevA.33.1141.  Google Scholar

[20]

M. Kesseböhmer, Large deviation for weak Gibbs measures and multifractal spectra, Nonlinearity, 14 (2001), 395-409.  doi: 10.1088/0951-7715/14/2/312.  Google Scholar

[21]

K. M. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2,171, Amer. Math. Soc., Providence, RI (1996), 107–140. doi: 10.1090/trans2/171/10.  Google Scholar

[22]

Y. Kifer, Equilibrium states for random expanding transformations, Random Comput. Dynam., 1 (1992/93), 1-31.   Google Scholar

[23]

Y. Kifer, Fractals via random iterated function systems and random geometric constructions, in Fractal geometry and stochastics (Finsterbergen, 1994), Progr. Probab., Birkhäuser, Basel, 37 (1995), 145–164. doi: 10.1007/978-3-0348-7755-8_7.  Google Scholar

[24]

Y. Kifer, Fractal dimensions and random transformations, Trans. Amer. Math. Soc., 348 (1996), 2003-2038.  doi: 10.1090/S0002-9947-96-01608-X.  Google Scholar

[25]

Y. Kifer, On the topological pressure for random bundle transformations, in Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 202 (2001), 197–214. doi: 10.1090/trans2/202/14.  Google Scholar

[26]

Y. Kifer, Thermodynamic formalism for random transformations revisited, Stoch. Dyn., 8 (2008), 77-102.  doi: 10.1142/S0219493708002238.  Google Scholar

[27]

Y. Kifer and P. -D. Liu, Random dynamics, in Handbook of dynamical systems, Elsevier B. V., Amsterdam, 1B (2006), 379–499. doi: 10.1016/S1874-575X(06)80030-5.  Google Scholar

[28]

S. P. Lalley and D. Gatzouras, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J., 41 (1992), 533-568.  doi: 10.1512/iumj.1992.41.41031.  Google Scholar

[29]

K.-S. Lau and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math., 141 (1999), 45-96.  doi: 10.1006/aima.1998.1773.  Google Scholar

[30]

N. Luzia, A variational principle for the dimension for a class of non-conformal repellers, Ergod. Th. & Dynam. Sys., 26 (2006), 821-845.  doi: 10.1017/S0143385705000659.  Google Scholar

[31]

J.-H. MaZ.-Y. Wen and J. Wu, Besicovitch subsets of self-similar sets, Ann. Inst. Fourier, 52 (2002), 1061-1074.  doi: 10.5802/aif.1911.  Google Scholar

[32]

J.-H. Ma and Z.-Y. Wen, Hausdorff and Packing measure of sets of generic points: A ZeroInfinity Law, J. London Math. Soc., 69 (2004), 383-406.  doi: 10.1112/S0024610703005040.  Google Scholar

[33]

P. T. Maker, The ergodic theorem for a sequence of functions, Duke Math. J., 6 (1940), 27-30.  doi: 10.1215/S0012-7094-40-00602-0.  Google Scholar

[34]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[35]

V. Mayer, B. Skorulski and M. Urbański, Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry Lecture Notes in Mathematics, 2036, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-23650-1.  Google Scholar

[36]

E. Olivier, Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures, Nonlinearity, 12 (1999), 1571-1585.  doi: 10.1088/0951-7715/12/6/309.  Google Scholar

[37]

L. Olsen, A multifractal formalism, Adv. Math., 116 (1995), 82-196.  doi: 10.1006/aima.1995.1066.  Google Scholar

[38]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.  doi: 10.1016/j.matpur.2003.09.007.  Google Scholar

[39]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. Ⅳ: Divergence points and packing dimension, Bull. Sci. Math., 132 (2008), 650-678.  doi: 10.1016/j.bulsci.2008.08.002.  Google Scholar

[40]

N. Patzschke, Self-Conformal Multifractal Measures, Adv. Appl. Math., 19 (1997), 486-513.  doi: 10.1006/aama.1997.0557.  Google Scholar

[41]

Y. 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), 105-153.  doi: 10.1007/BF02506387.  Google Scholar

[42]

Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys., 86 (1997), 233-275.  doi: 10.1007/BF02180206.  Google Scholar

[43]

D. A. Rand, The singularity spectrum f(α) for cookie-cutters, Ergod. Th. & Dynam. Sys., 9 (1989), 527–541. doi: 10.1017/S0143385700005162.  Google Scholar

[44]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Th. & Dynam. Sys., 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820.  Google Scholar

[45]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Hyperbolic dynamics, fluctuations and large deviations, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 89 (2015), 81-117.  doi: 10.1090/pspum/089/01485.  Google Scholar

[46]

L. Shu, The multifractal analysis of Birkhoff averages for conformal repellers under random perturbations, Monatsh. Math., 159 (2010), 81-113.  doi: 10.1007/s00605-009-0149-4.  Google Scholar

Figure 1.  Choice of $\overline\kappa_1$
Figure 2.  Choice of $\overline\kappa_{i+1}$
Figure 3.  A cover for $B(x, r)\cap X_{\omega}$
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