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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

[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.

[43]

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

[44]

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

[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.

[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.

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

[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.

[43]

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

[44]

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

[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.

[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.

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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