# American Institute of Mathematical Sciences

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:

show all references

##### References:
Choice of $\overline\kappa_1$
Choice of $\overline\kappa_{i+1}$
A cover for $B(x, r)\cap X_{\omega}$
 [1] Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303 [2] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [3] Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 [4] Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699 [5] Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006 [6] Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017 [7] 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 [8] Michael Scheutzow, Maite Wilke-Berenguer. Random Delta-Hausdorff-attractors. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1199-1217. doi: 10.3934/dcdsb.2018148 [9] Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647 [10] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [11] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [12] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [13] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [14] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [15] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [16] Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks & Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013 [17] Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 [18] Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1 [19] Nuno Luzia. Measure of full dimension for some nonconformal repellers. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 291-302. doi: 10.3934/dcds.2010.26.291 [20] Antoine Gournay. A dynamical approach to von Neumann dimension. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 967-987. doi: 10.3934/dcds.2010.26.967

2018 Impact Factor: 1.143