Stable sets, hyperbolicity and dimension

Previous Article
The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation
DCDS Home
This Issue
Next Article
Invariant criteria for existence of bounded positive solutions

April 2005, 12(3): 403-412. doi: 10.3934/dcds.2005.12.403

Stable sets, hyperbolicity and dimension

Rasul Shafikov ^1, and Christian Wolf ^2,

1.	Department of Mathematics, Middlesex College, The University of Western Ontario, London, Ontario N6A 5B7, Canada
2.	Department of Mathematics, Wichita State University, Wichita, Kansas, 67260, United States

Received September 2003 Revised August 2004 Published December 2004

Abstract
Related Papers

In this note we derive an upper bound for the Hausdorff and box dimension of the stable and local stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that dim$_H W^s(\Lambda)=n$ is equivalent to the existence of a SRB-measure. We also discuss related results for expanding maps.

Keywords: box dimension, stable sets., SRB measure, Hausdorff dimension, Hyperbolic sets.

Mathematics Subject Classification: 37C45, 37C40, 37DX.

Citation: Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403

[1]	Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[2]	Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.
[3]	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
[4]	Boris Hasselblatt and Jorg Schmeling. Dimension product structure of hyperbolic sets. Electronic Research Announcements, 2004, 10: 88-96.
[5]	Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
[6]	Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015
[7]	Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[8]	Todd Young. Partially hyperbolic sets from a co-dimension one bifurcation. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 253-275. doi: 10.3934/dcds.1995.1.253
[9]	Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254
[10]	Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553
[11]	Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125
[12]	Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435
[13]	V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117
[14]	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
[15]	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
[16]	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
[17]	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
[18]	Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[19]	Todd Fisher. Hyperbolic sets with nonempty interior. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 433-446. doi: 10.3934/dcds.2006.15.433
[20]	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

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences