American Institute of Mathematical Sciences

Advanced
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

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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254
[10]

Lisa Hernandez Lucas. Properties of sets of subspaces with constant intersection dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052
[11]

Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553
[12]

Serafin Bautista, Yeison Sánchez. Sectional-hyperbolic Lyapunov stable sets. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2011-2016. doi: 10.3934/dcds.2020103
[13]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125
[14]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435
[15]

V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117
[16]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[17]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[18]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405
[19]

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
[20]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy